A new fractal dimension: The topological Hausdorff dimension

A NEW FRA CT AL DIMENSION: THE TOPOLOGICAL HA USDORFF DIMENSION RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKES Abstract. W e int ro duce a new concept of dimension for metric spaces, the so-called top olo gica l Hausdorﬀ dimension . It is deﬁn ed b y a v ery natural com bination of the deﬁnitions of the to p ological dimension and the Hausdorﬀ dimension. The v alue of th e topological Hausdorﬀ dimension is alw a ys betw een the top ological dimension and the Hausdorﬀ dim ensi on, in particular, th is new dimension is a non-trivial low er estimate for the Hausdorﬀ dimension. W e examine the basic properties of this ne w notion of di mension, compare it to other w ell-known notions, determine its v alue f or some classical fr actals suc h as the Sierpinski carpet, the von Koc h sno wﬂak e curve, Kak eya sets, the trail of the Brownian motion, etc. As our ﬁrst application, we generalize the celebrated result of Cha ye s, Cha ye s and Durrett about the phase transition of the connectedness of the limit set o f Mandelbrot’s fractal p ercolation pr o cess. They prov ed that certain curv es show up in the l imit set when passing a cr i tical probability , and we prov e that actually ‘thick’ families of curve s show up, where roughly sp eak- ing the word thic k means that the curves can b e parametrized in a natural wa y by a set of large Hausdorﬀ dim ension. The pro of of this is basically a low er estimate of the topol ogical Hausdorﬀ dimension of the li mit set. F or the sak e of completeness, we also give an upp er estimate and conclude that in the non-trivial cases the top ological H ausdorﬀ di mension is al most s urely strictly below the Hausdorﬀ dimension. Finally , as our second application, w e show that the top ological H ausdorﬀ dimension i s precisely the right notion to describ e the Hausdorﬀ di mension of the leve l sets of the generic contin uous function (in the s ense of Bair e cat egory) deﬁned on a compact metric space. Contents 1. Int ro duction 2 2. Preliminarie s 6 3. Equiv alent deﬁnitions of the to polo gical Hausdor ﬀ dimension 6 4. Basic prop erties of the top ologica l Hausdorﬀ dimensio n 9 5. Calculating the to polo gical Hausdor ﬀ dimension 14 2010 Mathematics Subj ect Classiﬁc ation. Primary: 28A78, 28A80; Seconda ry: 54F45, 60J65, 60K35. Key wor ds and phr ases. Hausdorﬀ dimension, top ological H ausdorﬀ dimension, Brownian mo- tion, M andelbrot’s fractal p ercolation, critical probabili ty , l ev el sets, generic, ty pical con tin uous functions, f r actals. The ﬁrst author was supp orted by the Hungarian Scientiﬁc Research F und gran t no. 72655. The second author w as supp orted by the Hungarian Scien tiﬁc Researc h F und grants no. K075242 and K104178. The third author was supp orted by the Hungarian Scientiﬁc Research F und gran ts no. 72655, 61600, and 83726. 1 2 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S 5.1. Some cla ssical fracta ls 14 5.2. Kakey a sets 15 5.3. Brownian motio n 17 6. Application I: Mandelbr ot’s fracta l p ercolation pr o cess 23 6.1. Prepara tio n 24 6.2. Pro of o f Theorem 6.2; the low e r estimate of dim tH M 26 6.3. The upp er estimate of dim tH M 31 7. Application I I: The Hausdorﬀ dimension o f the le vel sets of the gener ic contin uo us function 32 7.1. Arbitrary compact metr ic spaces 33 7.2. Homogeneous and self- s imilar compact metric spaces 35 8. Op en P roblems 37 Ac knowledgmen ts 38 References 38 1. Introduction The term ‘fractal’ was intro duced b y Mandelbrot in his celebrated b o ok [20]. He formally deﬁned a subset o f a Euc lide a n space to b e a fr actal if its top olo g ical dimension is strictly smaller than its Hausdorﬀ dimensio n. T his is just one exa mple to illustrate the fundamental role these tw o notions of dimension play in the study of fractal sets. T o mention another s uc h example, let us recall that the top ologic al dimension of a metric space X is the inﬁmum of the Haus dorﬀ dimensio ns o f the metric spa ces homeomo rphic to X , see [12]. The main goal o f this pap er is to introduce a new concept of dimensio n, the so - called t op olo gic al Hausdorﬀ dimension , that interp olates the tw o a bove mentioned dimensions in a very natural way . Let us r ecall the deﬁnition o f the (small inductive) top ological dimensio n (see e.g. [6, 12]). Deﬁnition 1.1 . Set dim t ∅ = − 1 . The top olo gic al dimension o f a non-empty metric space X is deﬁned by induction as dim t X = inf { d : X has a basis U such that dim t ∂ U ≤ d − 1 for every U ∈ U } . Our new dimensio n will b e deﬁned a nalogously , how ev er, note that this second deﬁnition will not b e inductive, a nd also tha t it can attain no n-in teger v alues as well. The Hausdorﬀ dimension of a metric spac e X is denoted by dim H X , see e.g. [7] or [21]. In this pap er we adopt the conven tion that dim H ∅ = − 1. Deﬁnition 1.2. S et dim tH ∅ = − 1. The top olo gic al Haus dorﬀ dimension of a non-empty metric space X is deﬁned as dim tH X = inf { d : X has a basis U such that dim H ∂ U ≤ d − 1 for every U ∈ U } . (Both notions o f dimension ca n attain the v a lue ∞ a s well, actually we use the conv en tion ∞ − 1 = ∞ , hence d = ∞ is a mem b er of the ab ov e set.) F ro m now on generic is always understo o d in the s ense of Bair e category . It was not this analog y that initiated the study of this new concept. Over the last 30 years there has b een a la rge interest in studying dimensio ns o f v a rious sets A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 3 related to generic c ont inuous maps. Mauldin and Williams [22] prov ed that the Hausdorﬀ dimensio n o f the gra ph o f the generic f ∈ C [0 , 1] is 1 , while Humke and Petrusk a [11] show ed that its packing dimension is 2. The b ox dimensions of graphs o f generic contin uous functions were in vestigated by Hyde, Laschos, Ols en, Petrykiewicz, and Shaw [13]. Balk a , F a r k as, F ra s er and Hyde [2] pr oved that for the generic contin uous map f : K → R n we hav e dim H f ( K ) = min { dim t K, n } for all compact metr ic s paces K . Bruckner a nd Ga rg [4] describ ed the level set str uctur e of the generic f ∈ C [0 , 1] from the topo logical p oint of view. F rom the metric p oint of view, it is well-known that each non-empty level set of the gene r ic f ∈ C [0 , 1] is of Haus dorﬀ dimensio n 0. Kirchheim [15] consider ed the level sets of the gener ic contin uo us function f : [0 , 1] d → R . He prov ed that for every y ∈ in t f ([0 , 1] d ) we hav e dim H f − 1 ( y ) = d − 1, that is, a s o ne would exp ect, ‘most’ level se ts are o f Hausdorﬀ dimension d − 1. The next problem is ab out ge neralizations of this r e s ult to frac ta l sets in pla ce of [0 , 1] d , this is where our origina l motiv ation came fro m. Problem 1.3. D escrib e the Hausdorﬀ dimension of the level sets of the generic c ontinuou s function deﬁne d on a c omp act metric sp ac e. It has turned out that the top ologica l Hausdorﬀ dimensio n is the right concept to deal with this problem. W e will essentially prove that the v alue d − 1 in Kir c hheim’s result has to b e r eplaced b y dim tH K − 1, see the end of this int ro duction or Section 7 for the details. W e would als o like to ment ion ano ther p otentially very int eresting motiv ation of this new concept. Unlike most well-known notio ns o f dimension, such as packing or box-counting dimensions, the top olog ical Hausdo rﬀ dimensio n is smaller than the Hausdorﬀ dimension. As it is often an imp or ta n t and diﬃcult ta sk to estimate the Hausdorﬀ dimension from b elow, this gives another r eason to study the to polo gical Hausdorﬀ dimens io n. It is also worth mentioning that there is another recent appr o ach by Urba ´ nski [31] to combine the top olog ic al dimension and the Hausdorﬀ dimension. How ever, his new concept, ca lle d the trans ﬁnite Haus dorﬀ dimens io n is q uite diﬀerent in nature fro m ours, e.g. it takes ordina l n umbers a s v a lues. Moreov er, the ﬁrst listed author (using ideas of U. B . Dar ji and the third listed author) recently g eneralized the r e s ults of the pap er for ma ps taking v alues in R n instead of R . The new conce pt of dimensio n needed to describ e the Hausdorﬀ dimension of the ﬁb ers of the g eneric contin uous map is called the n th inductive top ological Haus do rﬀ dimension, see [3 ]. Next we sa y a few w ords ab out the main r e sults and the org anization of the pap er. In Section 3 we discuss some alterna tiv e deﬁnitions of the top ologica l Hausdo rﬀ dimension yielding the same concept. Recall that the following cla s sical theor em in fact describ es an alterna tive recursive deﬁnition of the top ologic al dimension. Theorem 3.1. If X is a non-empty sep ar able metric sp ac e then dim t X = min { d : ∃ A ⊆ X such that dim t A ≤ d − 1 and dim t ( X \ A ) ≤ 0 } . 4 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S The next res ult shows tha t by repla cing one ins tance o f dim t A by dim H A we again obta in the notion of top ological Haus do rﬀ dimension. Theorem 3.6. If X is a non-empty sep ar able metric sp ac e then dim tH X = min { d : ∃ A ⊆ X s u ch that dim H A ≤ d − 1 and dim t ( X \ A ) ≤ 0 } . As a co rollary we also obtain that a ctually inf = min in our or iginal deﬁnition of the topo logical Hausdorﬀ dimensio n, which is not o nly int eresting, but will also b e used in one o f the applications . W e dis cuss the ana logues of the other deﬁnitions of the top ological dimension as well, s uch as the large inductiv e dimension and the Leb esgue covering dimension. In Sectio n 4 we in vestigate the basic prop erties of the top ologica l Hausdor ﬀ dimension. Among others, we prov e the following. Theorem 4.4. dim t X ≤ dim tH X ≤ dim H X . W e also verify that dim tH X satisﬁes some sta ndard pro per ties of a dimension, such as mo notonicity , bi-Lipschitz inv ar iance and co untable stability for closed sets. W e discuss the existence of G δ hu lls and F σ subsets with the s ame to polo gical Hausdorﬀ dimensio n, a s well. Mo reov er, we chec k that this concept is genuinely new in that dim tH X ca nnot b e expressed as a function of dim t X a nd dim H X . In Sec tion 5 we co mpute dim tH X for so me cla ssical fracta ls, like the Sierpi ´ nski triangle and ca rpe t, the von Ko ch c urve, etc. F or example Theorem 5.4. L et T b e the Sierpi´ nski c arp et. Then dim tH T = log 6 log 3 = log 2 log 3 + 1 . (Note that dim t T = 1 and dim H T = log 8 log 3 while the Ha usdorﬀ dimensio n of the triadic Ca n tor set equals log 2 log 3 .) W e a lso consider K akey a s ets (see [7] or [21]). Unfortunately , our metho ds do not give any useful information concerning the K akey a Co njecture. Theorem 5.6. F or every d ∈ N + ther e exist a c omp act Kakeya set of top olo gic al Hausdorﬀ dimension 1 in R d . F ollowing K¨ or ner [17] we prove somewha t more, since we essentially show that the gener ic element of a car efully chosen space is a Ka k eya set o f to polo gical Haus- dorﬀ dimensio n 1. W e show that the ra ng e of the Brownian motio n a lmost surely (i.e. with proba- bilit y 1) has top olog ic al Hausdorﬀ dimension 1 in every dimensio n exc ept p erhaps 2 and 3. These tw o cas es remain the mos t intriguing op en problems of the pap er. Problem 5.8. Det ermine t he almost sur e top olo gic al Hausdorﬀ dimension of t he r ange of the d -dimensional Br ownian motion for d = 2 or 3 . Equivalently, de- termine the smal lest c ≥ 0 such that the r ange c an b e de c omp ose d int o a total ly disc onne ct e d set and a set of Hau s dorﬀ dimension at most c − 1 almost sur ely. W e also rela te the pla na r case to a well-known op en pr o blem of W. W erner and solve the dual version of this problem, that is, the version in w hich the notion of Wiener measure is replaced by Bair e ca teg ory . In a simila r vein, we also show that A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 5 the rang e of the generic co n tinuous map f : [0 , 1] → R d is of top o logical Hausdorﬀ dimension 1 for every d . As our ﬁrst applicatio n in Section 6 w e g eneralize a res ult o f Chayes, Chayes and Durrett abo ut the phase transitio n of the connectedness of the limit set of Mandelbrot’s fracta l p ercola tion pr o cess. T his limit s et M = M ( p,n ) is a rando m Cantor set, which is c onstructed b y dividing the unit squar e into n × n equal sub-square s and keeping e ach of them indep endently with probability p , and then rep eating the same pro cedur e recurs iv ely for e very sub-squa re. (See Section 6 for more details.) Theorem 6. 1 (Chay es-Chay es-Durrett, [5]) . Ther e exists a critic al pr ob ability p c = p ( n ) c ∈ (0 , 1) su ch that if p p c then M c ontains a nontrivial c onne cte d c omp onent with p ositive pr ob ability. It will b e easy to see that this theor em is a sp ecial ca se of our nex t re s ult. Theorem 6. 2. F or every d ∈ [0 , 2) t her e exists a critic al pr ob ability p ( d ) c = p ( d,n ) c ∈ (0 , 1 ) such that if p p ( d ) c then dim tH M > d almost sur ely (pr ovide d M 6 = ∅ ). Theorem 6.1 ess e ntially says that certain cur v es show up at the c r itical proba- bilit y , and our pro of will show that even ‘thick’ families of curves show up, which roughly sp eaking means a ‘Lipschitz-lik e copy’ of C × [0 , 1] with dim H C > d − 1. W e also give a numerical upper b ound for dim tH M which implies the following. Corollary 6.18. Almost sur ely dim tH M < dim H M or M = ∅ . In Sec tion 7 we answer Problem 1.3 as follows. Corollary 7.14. If K is a c omp act metric sp ac e with dim t K > 0 then for the generic f ∈ C ( K ) sup  dim H f − 1 ( y ) : y ∈ R  = dim tH K − 1 . (It is well-known that if dim t K = 0 then the generic f ∈ C ( K ) is one-to -one, th us every non-empty level set is of Hausdorﬀ dimensio n 0, see [1, Lemma 2.6] for a pro o f.) If K is also suﬃciently homog eneous, e.g. self-similar then we can actually say more. Corollary 7.17. L et K b e a self-similar c omp act metric sp ac e with dim t K > 0 . Then for the generic f ∈ C ( K ) for t he generic y ∈ f ( K ) dim H f − 1 ( y ) = dim tH K − 1 . It can actually also b e shown that the supremum is a ttained in Corollary 7.14. On the other hand, one ca nno t say more in a sens e , sinc e there is a K ⊆ R 2 such that for the gener ic f ∈ C ( K ) there is a unique y ∈ R for which dim H f − 1 ( y ) = dim tH K − 1. Moreov er, in certain situations we can replace ‘the generic y ∈ f ( K )’ with ‘for every y ∈ int f ( K )’ as in Kirchheim’s theorem. The results of this la st pa ragra ph app eared elsewher e , see [1]. Finally , in Section 8 we list some o pen problems. 6 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S 2. Preliminaries Let ( X , d ) b e a metric space. F or A, B ⊆ X let us deﬁne dist( A, B ) = inf { d ( x, y ) : x ∈ A, y ∈ B } . Let B ( x, r ) and U ( x, r ) sta nd for the closed and o pen ba ll of radius r centered a t x , resp ectively . Set B ( A, r ) = { x ∈ X : dist ( { x } , A ) ≤ r } and U ( A, r ) = { x ∈ X : dist( { x } , A ) < r } . W e denote by cl A , int A and ∂ A the closur e, int erior and b oundary of A , resp ectively . The dia meter o f a se t A is denoted by diam A . W e use the conv e n tion diam ∅ = 0. F or tw o metric s pa ces ( X , d X ) and ( Y , d Y ) a function f : X → Y is Lipschitz if there e x ists a co nstant C ∈ R such that d Y ( f ( x 1 ) , f ( x 2 )) ≤ C · d X ( x 1 , x 2 ) for all x 1 , x 2 ∈ X . The smallest such co nstant C is called the Lips c hitz constant of f and denoted by Lip( f ). A function f : X → Y is ca lled bi-Lipschitz if f is a bijection and b oth f a nd f − 1 are Lipschitz. F or a metric spa ce X and s ≥ 0 the s -dimensional Hausdorﬀ me asu re is deﬁned as H s ( X ) = lim δ → 0+ H s δ ( X ), wher e H s δ ( X ) = inf ( ∞ X i =1 (diam U i ) s : X ⊆ ∞ [ i =1 U i , ∀ i diam U i ≤ δ ) . Recall that H s ∞ ( X ) = inf ( ∞ X i =1 (diam U i ) s : X ⊆ ∞ [ i =1 U i ) is the s -dimensional Hausdorﬀ c ontent . The Hausdorﬀ dimension of X is deﬁned a s dim H X = inf { s ≥ 0 : H s ( X ) = 0 } . It is not diﬃcult to see using the r egularity of H s δ that every set is c o n tained in a G δ set o f the s ame Hausdor ﬀ dimension. F or more info r mation on these concepts see [7] or [21]. Let X be a c omplete metric space. A set is somewher e dense if it is dense in a non-empt y op en s et, and otherwise it is ca lled nowher e dense . W e sa y tha t M ⊆ X is me ager if it is a countable union of no where dense s ets, and a s et is called c o-me ager if its complement is meager. By Baire’s Categor y Theo rem c o - meager sets are dens e. It is not diﬃcult to show tha t a s et is co -meager iﬀ it contains a dense G δ set. W e say that the generic element x ∈ X has pro p erty P if { x ∈ X : x ha s prop erty P } is co -meager. The term ‘typical’ is also us e d instead of ‘gener ic’. See e .g. [14] fo r more on these concepts. A top olo gical spa ce X is ca lled total ly disc onne ct e d if X is empty or every c o n- nected comp onent of X is a singleto n. If dim t X ≤ 0 then X is c le a rly totally disconnected. If X is lo cally co mpact then dim t X ≤ 0 iﬀ X is totally disc o n- nected, see [6, 1.4.5 .]. If X is a σ -compac t metric spac e then dim t X ≤ 0 iﬀ X is totally disconnected, se e the countable stability of to p olo gical dimensio n zero for closed s ets [6, 1.3.1.]. 3. Equiv al ent definitions of the topological Hausdorff dimension The g oal of this section is to prov e so me equiv ale n t deﬁnitions which will play an imp o rtant r ole later. Perhaps the main p oint here is that while the original deﬁnition is of lo cal nature, we c a n ﬁnd an equiv a len t globa l deﬁnition. Let us recall the following deco mp ositio n theorem for the top olo g ical dimension, see [6, 1.5.7 .], which can b e rega rded as a n equiv a len t deﬁnition. A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 7 Theorem 3.1 . If X is a non-empty sep ar able metric sp ac e then dim t X = min { d : ∃ A ⊆ X such that dim t A ≤ d − 1 and dim t ( X \ A ) ≤ 0 } . (Note that as ab ov e, ∞ is a ssumed to b e a member of the a bove set, moreover we use the conv ention min { ∞} = ∞ .) The main goa l of this section is to prove Theo rem 3.6, an analog ous decomp o- sition theor em for the top ological Haus do rﬀ dimension, which yields a n eq uiv alent deﬁnition o f the top ologic al Hausdorﬀ dimens ion. As a by-pro duct, we o btained that in the deﬁnition of dim tH X the inﬁmum is attained, see Corollar y 3.8. Remark 3.2. O ne can actually chec k that, as usual in dimension theory , the assumption o f separa bilit y cannot be dr oppe d. Let X be a given non-empty separa ble metric space . F or the sake of notational simplicity we us e the following notation. Notation 3. 3 . P tH = { d : X has a basis U such that dim H ∂ U ≤ d − 1 for every U ∈ U } , P dH = { d : ∃ A ⊆ X suc h that dim H A ≤ d − 1 and dim t ( X \ A ) ≤ 0 } . W e ass ume ∞ ∈ P tH , P dH . The following lemmas are the heart of the section. Lemma 3.4. P tH = P dH . Pr o of. Fir st we prov e P tH ⊆ P dH . Assume d ∈ P tH and d < ∞ . Then there exists a countable basis U of X such that dim H ∂ U ≤ d − 1 for all U ∈ U . Let A = S U ∈U ∂ U , then the co un table stability of the Hausdorﬀ dimension yields dim H A ≤ d − 1, and the deﬁnition o f A clearly implies dim t ( X \ A ) ≤ 0. Hence d ∈ P dH . Now we prov e P dH ⊆ P tH . Assume d ∈ P dH and d < ∞ . Let us ﬁx x ∈ X and r > 0. T o verify d ∈ P tH we need to ﬁnd an op en set U ⊆ X such that x ∈ U ⊆ U ( x, r ) a nd dim H ∂ U ≤ d − 1. Since d ∈ P dH , there is a set A ⊆ X suc h that dim H A ≤ d − 1 and dim t ( X \ A ) ≤ 0 . As X \ A is a separa ble subspa ce of X with top ologica l dimension 0, by the separ a tion theorem for top ologica l dimension zero [6, 1.2 .11.] there is a s o-called partition b et ween x and X \ U ( x, r ) dis join t from X \ A . This means that there exist disjoint o pen sets U, U ′ ⊆ X such that x ∈ U , X \ U ( x, r ) ⊆ U ′ and ( X \ ( U ∪ U ′ )) ∩ ( X \ A ) = ∅ . In particula r , x ∈ U ⊆ U ( x, r ). Moreov er, ∂ U ∩ ( X \ A ) = ∅ , s o ∂ U ⊆ A , thus dim H ∂ U ≤ dim H A ≤ d − 1. Hence d ∈ P tH .  Lemma 3.5. inf P dH ∈ P dH . Pr o of. L e t d = inf P dH , we may assume d < ∞ . Set d n = d + 1 /n for all n ∈ N + . As d n ∈ P dH , ther e exist se ts A n ⊆ X such that dim H A n ≤ d n − 1 and dim t ( X \ A n ) ≤ 0. W e may assume that the sets A n are G δ , since we ca n take G δ hu lls with the same Hausdorﬀ dimension. Let A = T ∞ n =1 A n , then cle a rly dim H A ≤ d − 1. As X \ A n are F σ sets such that dim t ( X \ A n ) ≤ 0 and X \ A = S ∞ n =1 ( X \ A n ), countable stability of topolo g ical dimens io n zero for F σ sets [6, 1.3.3 . Corollary] yields dim t ( X \ A ) ≤ 0. Hence d ∈ P dH .  Now the main result of the section is an eas y consequence of Lemmas 3.4, 3.5 and the deﬁnition of the top ologica l Hausdorﬀ dimension. 8 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S Theorem 3.6 . If X is a non-empty sep ar able metric sp ac e then dim tH X = min { d : ∃ A ⊆ X s u ch that dim H A ≤ d − 1 and dim t ( X \ A ) ≤ 0 } . Since every s e t is contained in a G δ set of the s ame Haus do rﬀ dimension, and a σ - compact metric space is totally disconnected iﬀ it ha s top olo gical dimension zero, the ab ov e theorem yields the following eq uiv alent deﬁnition. Theorem 3.7 . F or a non-empty σ -c omp act metric sp ac e X dim tH X = min { d : ∃ A ⊆ X s u ch that dim H A ≤ d − 1 and X \ A is total ly disc onne cte d } . Moreov er, as a by-pro duct of Lemma 3.4 and Lemma 3.5 w e obtain that the inﬁm um is attained in the o r iginal deﬁnition, which will play a role in one of the applications. Corollary 3.8. If X is a non-empty sep ar able m et ric sp ac e t hen dim tH X = min { d : X has a b asis U such that dim H ∂ U ≤ d − 1 for every U ∈ U } . Remark 3.9 . There are even mor e equiv alent deﬁnitions of the top ologica l dimen- sion, and one can prove that the a ppropriate analo gues r e s ult in the same notion as well, but since these statements will not be used in the sequel we only state the results her e. F or some details consult [3]. Moreov er, Section 7 contains further equiv alent deﬁnitions in the c o mpact case in connection with the level sets of generic contin uous functions. The next clas sical theor e m shows that the notion o f lar ge inductive dimension coincides with the other deﬁnitions of top olo gical dimension for separable metric spaces. Theorem 3.1 0. F or every non-empty sep ar able m et ric sp ac e X dim t X = min { d : ∀ F ⊆ X close d and ∀ V op en with F ⊆ V , ∃ U op en such t hat F ⊆ U ⊆ V and dim t ∂ U ≤ d − 1 } . The natural analogue y ields the same concept a gain. Theorem 3.1 1. F or every non-empty sep ar able m et ric sp ac e X dim tH X = min { d : ∀ F ⊆ X close d and ∀ V op en with F ⊆ V , ∃ U op en such that F ⊆ U ⊆ V and dim H ∂ U ≤ d − 1 } . Next we take up the Leb esgue covering dimension. F or a family A of sets and m ∈ N + let T m ( A ) denote the set of p oints cov ered by a t least m member s of A . Theorem 3.1 2. F or every non-empty sep ar able m et ric sp ac e X dim t X = min { d : ∀ ﬁ n ite op en c over U of X ∃ a ﬁnite op en r eﬁnement V of U s u ch that T d +2 ( V ) = ∅} . The analogo us re s ult is the following. Theorem 3.1 3. F or every sep ar able metric sp ac e X with dim t X > 0 dim tH X = min { d : ∀ ε > 0 ∀ ﬁnite op en c over U of X ∃ a ﬁnite op en r eﬁnement V of U s u ch that H d − 1+ ε ∞ ( T 2 ( V )) ≤ ε } . A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 9 4. Basic proper ties of the topological H ausdorff dimension The ma in go al of this section is to investigate the ba sic pr op e rties of the top o- logical Hausdo rﬀ dimensio n. It will turn out that some pro per ties of this dimension are metric in nature, while o thers ar e top olo gical. This enhances the philoso phy that our new notion is really ‘b etw een’ the top olo gical and Ha usdorﬀ dimensions . Let X b e a metr ic s pa ce. Since dim t X = − 1 ⇐ ⇒ X = ∅ ⇐ ⇒ dim H X = − 1 , we ea sily obtain F act 4 .1. dim tH X = 0 ⇐ ⇒ dim t X = 0 . As dim H X is either − 1 or at least 0 , we obtain F act 4.2. The t op olo gic al Hausdorﬀ dimension of a non- empty sp ac e is either 0 or at le ast 1 . These tw o facts easily yield Corollary 4. 3. Every metric sp ac e with a non- t rivial c onne cte d c omp onent has top olo gic al Hausdorﬀ dimension at le ast one. The next theorem states that the topolo gical Hausdorﬀ dimension is betw een the to p olo gical and the Ha us dorﬀ dimension. Theorem 4.4 . F or every metr ic sp ac e X dim t X ≤ dim tH X ≤ dim H X . Pr o of. W e can clear ly assume that X is non-empty . It is well-known that dim t X ≤ dim H X (see e.g. [1 2]), which easily implies dim t X ≤ dim tH X using the deﬁni- tions. The second ineq ua lit y is obvious if dim H X = ∞ . If dim H X < 1 then dim t X = 0 (since dim t X ≤ dim H X and dim t X only takes int eger v alues) and by F act 4.1 we o bta in dim tH X = 0, hence the second inequality holds. Therefor e we ma y assume that 1 ≤ dim H X < ∞ . The following lemma is basically [21, Thm. 7.7]. I t is only sta ted there in the sp ecial case X = A ⊆ R n , but the pro of works v erbatim for all metric spaces X . Lemma 4. 5 . L et X b e a metric sp ac e and f : X → R m b e Lipschitz. If s ≥ m , then (4.1) Z ⋆ H s − m  f − 1 ( y )  d H m ( y ) ≤ c ( m ) Lip( f ) m H s ( X ) , wher e R ⋆ denotes the upp er inte gr al and c ( m ) is a ﬁnite c onstant dep ending only on m . Now we r eturn to the pro of of Theorem 4.4. W e ﬁx x 0 ∈ X and deﬁne f : X → R by f ( x ) = d X ( x, x 0 ). Using the triangle inequality it is easy to see that f is Lipschitz with Lip( f ) ≤ 1 . W e ﬁx n ∈ N + and apply Le mma 4 .5 for f and s = dim H X + 1 n > 1 = m . Hence Z ⋆ H s − 1 ( f − 1 ( y )) d H 1 ( y ) ≤ c (1) H s ( X ) = 0 . Thu s H s − 1 ( f − 1 ( y )) = H dim H X + 1 n − 1 ( f − 1 ( y )) = 0 holds for a.e. y ∈ R . Since this is true for all n ∈ N + , we obtain that dim H f − 1 ( y ) ≤ dim H X − 1 for a.e. y ∈ R . F r om the deﬁnition of f it follows that ∂ U ( x 0 , y ) ⊆ f − 1 ( y ). Hence there is a neighbo rho o d 10 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S basis of x 0 with b oundaries of Hausdorﬀ dimension at most dim H X − 1, and this is true for all x 0 ∈ X , so there is a basis with b oundarie s o f Hausdorﬀ dimension at most dim H X − 1. By the deﬁnition o f the top ologica l Hausdor ﬀ dimensio n this implies dim tH X ≤ dim H X .  There are some elementary prop erties one exp ects from a notion of dimension. Now we verify some of these for the top olog ic al Hausdorﬀ dimens ion. Extension of the classical dimension. Theo rem 4 .4 implies that the top o- logical Hausdorﬀ dimensio n of a countable s et equa ls zer o, mor eov er, for op en subspaces of R d and for smo oth d -dimensional manifolds the top ologica l Hausdorﬀ dimension equa ls d . Monotonicity . Let X ⊆ Y . If U is a basis in Y then U X = { U ∩ X : U ∈ U } is a bas is in X , and ∂ X ( U ∩ X ) ⊆ ∂ Y U holds for all U ∈ U . This yields F act 4 .6 (Monotonicity) . If X ⊆ Y ar e metric sp ac es then dim tH X ≤ dim tH Y . Bi-Lipschitz i n v ariance. First we prove that the top ologica l Hausdor ﬀ dimen- sion do es not increa se under Lipschitz homeomorphisms. An ea sy consequence of this that our dimension is bi- Lipschit z inv ariant, and does not incre a se under an injectiv e Lipschitz map on a compa ct space. After obtaining corolla ries o f Theor em 4.7 we give so me examples illustrating the necessity of certain c o nditions in this theorem a nd its coro llaries. Theorem 4. 7. L et X , Y b e metric sp ac es. If f : X → Y is a Lipschitz home omor- phism then dim tH Y ≤ dim tH X . Pr o of. Since f is a homeomo rphism, if U is a basis in X then V = { f ( U ) : U ∈ U } is a basis in Y , and ∂ f ( U ) = f ( ∂ U ) for all U ∈ U . The Lipschitz prop erty of f implies that dim H ∂ V = dim H ∂ f ( U ) = dim H f ( ∂ U ) ≤ dim H ∂ U for a ll V = f ( U ) ∈ V . Thu s dim tH Y ≤ dim tH X .  Note that every bi-Lipschitz map and each injective contin uo us ma p deﬁned on a compact space is a homeo morphism. Therefore we obtain the following cor ollaries. Corollary 4.8 (Bi-Lipschitz inv ariance) . L et X , Y b e metric sp ac es. If f : X → Y is a bi-Lipschitz onto map, then dim tH X = dim tH Y . Corollary 4.9. If K is a c omp act metric sp ac e, and f : K → Y is one-to-one Lipschitz then dim tH f ( K ) ≤ dim tH K . The fo llowing example shows that we cannot dro p injectivity her e . Fir st we need a well-known lemma. Lemma 4.10. L et M ⊆ R b e m e asur able with p ositive L eb esgue me asur e. Then ther e ex ists a Lipschitz onto map f : M → [0 , 1] . Pr o of. L e t us choo s e a c o mpact s et C ⊆ M of p ositive Leb esgue measure. Deﬁne f : M → [0 , 1] by f ( x ) = λ (( − ∞ , x ) ∩ C ) λ ( C ) , where λ denotes the o ne-dimensional Leb esgue meas ure. Then it is not diﬃcult to see that f is Lipschitz (with Lip( f ) ≤ 1 λ ( C ) ) a nd f ( C ) = f ( M ) = [0 , 1].  A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 11 Example 4.11. Let K ⊆ R b e a Ca n tor s e t (that is, a set homeomorphic to the middle-thirds Cantor set) of p ositive Le besg ue mea s ure. By F act 4.1, dim tH K = dim t K = 0. Using Lemma 4.1 0 there is a Lipschitz map f : K → [0 , 1] such that f ( K ) = [0 , 1]. By Theorem 4.4, dim tH [0 , 1 ] = 1, hence dim tH K = 0 < 1 = dim tH [0 , 1 ] = dim tH f ( K ). The next example shows that Coro lla ry 4.9 do es not ho ld without the assumption of compac tness. W e even hav e a separa ble metric counterexample. Example 4. 12. Let C be the middle-thirds Cantor set, a nd f : C × C → [0 , 2 ] b e deﬁned b y f ( x, y ) = x + y . It is well-known a nd e a sy to see that f is Lipschitz and f ( C × C ) = [0 , 2]. Ther e fore o ne can sele c t a s ubs et X ⊆ C × C such that f | X is a bijectio n from X onto [0 , 2]. Then X is separ able metric. Mo notonicity and dim t ( C × C ) = 0 imply dim tH X ≤ dim tH ( C × C ) = 0. Therefore , f is o ne-to-one and Lipschitz on X but dim tH X = 0 < 1 = dim tH [0 , 2 ] = dim tH f ( X ). Our last example shows that the top olog ical Hausdo rﬀ dimension is not inv a riant under homeo morphisms. Not even for compact metric spaces. Example 4.13. Let C 1 , C 2 ⊆ R be Cantor s ets such that dim H C 1 6 = dim H C 2 . W e will see in Theorem 4.2 1 that dim tH ( C i × [0 , 1]) = dim H C i + 1 for i = 1 , 2. Hence C 1 × [0 , 1] and C 2 × [0 , 1] ar e homeo morphic co mpact metric spaces whos e top ological Haus do rﬀ dimensions disagr ee. Stability and counta ble stabili ty . As the following ex ample shows, similar ly to the case of top ological dimension, stabilit y do es not hold for non-close d s ets. That is, X = S k n =1 X n do es not imply dim tH X = max 1 ≤ n ≤ k dim tH X n . Example 4. 14. Theorem 4.4 implies dim tH R = 1 , and F act 4.1 yields dim tH Q = dim t Q = 0 and dim tH ( R \ Q ) = dim t ( R \ Q ) = 0. Thu s dim tH R = 1 > 0 = max { dim tH Q , dim tH ( R \ Q ) } , and therefo r e stability fails. As a cor ollary , we now show that as opp o sed to the c a se of Hausdorﬀ (and packing) dimension, ther e is no reaso nable family of meas ures inducing the top o- logical Hausdo rﬀ dimensio n. Let us say that a 1-par ameter family of mea sures { µ s } s ≥ 0 is monotone if µ s ( A ) = 0, s < t implies µ t ( A ) = 0. The family of Hausdorﬀ (or pa cking) measures certa inly satisﬁes this cr iterion. It is not diﬃ- cult to see that monotonicity implies that the induced notio n of dimensio n, that is, dim A = inf { s : µ s ( A ) = 0 } is countably stable. Hence we obtain Corollary 4. 1 5. Ther e is no monotone 1-p ar ameter family of me asur es { µ s } s ≥ 0 such that dim tH A = inf { s : µ s ( A ) = 0 } . How ever, just like in the case of top ologic a l dimension, even countable sta bility holds for close d sets. Theorem 4.16 (Countable stability for clos ed se ts) . L et X b e a sep ar able metric sp ac e and X = S n ∈ N X n , wher e X n ( n ∈ N ) ar e close d subsets of X . Then dim tH X = sup n ∈ N dim tH X n . Pr o of. Mo notonicity clear ly implies dim tH X ≥ sup n ∈ N dim tH X n . F or the oppo site inequa lit y let d = sup n ∈ N dim tH X n , we may assume d < ∞ . Theorem 3.6 yields that there ar e sets A n ⊆ X n such that dim H A n ≤ d − 1 and dim t ( X n \ A n ) ≤ 0 . W e may a ssume that the sets A n are G δ , since we can take G δ 12 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S hu lls with the same Hausdo rﬀ dimension. Let A = S ∞ n =0 A n ⊆ X , the co un table stability of the Hausdo r ﬀ dimension implies dim H A ≤ d − 1 . As the sets X n \ A n are F σ in X with dim t ( X n \ A n ) ≤ 0 and X \ A ⊆ S ∞ n =0 ( X n \ A n ), monotonicity and countable stability o f top ological dimension zero fo r F σ sets [6, 1.3 .3. Corolla ry] yield dim t ( X \ A ) ≤ 0. Finally , dim H A ≤ d − 1 and dim t ( X \ A ) ≤ 0 tog ether with Theorem 3.6 imply dim tH X ≤ d , and the pro of is complete.  Corollary 4.17 . Countable stability holds for F σ sets, as wel l. Regularity . Next we investigate the existence o f G δ hu lls and F σ subsets with the s a me top ologic a l Hausdorﬀ dimensio n. Theorem 4. 18 (Enlargement theore m for top ologica l Hausdorﬀ dimension) . If X is a metric sp ac e and Y ⊆ X is a sep ar able subsp ac e then ther e exists a G δ set G ⊆ X su ch that Y ⊆ G and dim tH G = dim tH Y . Pr o of. W e may assume Y 6 = ∅ . Theorem 3.6 implies that there exists Z ⊆ Y such that dim H Z = dim tH Y − 1 and dim t ( Y \ Z ) ≤ 0. Let A ⊆ X b e a G δ set such that Z ⊆ A and dim H A = dim H Z = dim tH Y − 1. By [6, 1.2.14 .] there exists a G δ set B ⊆ X s uch that Y \ Z ⊆ B and dim t B ≤ 0 . L et G = A ∪ B , then G is a G δ subset of X with Y ⊆ G . Since dim t ( G \ A ) ≤ dim t B ≤ 0, Theorem 3.6 implies that dim tH G ≤ dim tH A + 1 = dim tH Y , and mono to nicit y yields dim tH G = dim tH Y .  The following example shows tha t inner regularity do es not hold ev en for G δ subsets o f E uc lide a n spaces. Example 4.19. F or ev ery d ∈ N + Mazurkiewicz [23] constructed a G δ set G ⊆ [0 , 1 ] d +1 such that dim t G = d and G is totally disconnected (see [24, Thm. 3.9.3.] for a pro of in Eng lish). Theorem 4.4 implies that dim tH G ≥ dim t G = d . If F ⊆ G is clo sed in R d +1 then F is compact and tota lly disco nnected, thus dim t F ≤ 0, so F act 4.1 implies dim tH F ≤ 0 . Therefore countable stability of the top ologica l Hausdorﬀ dimensio n for closed sets yields that every F σ subset of G has top olog ic al Hausdorﬀ dimens io n at most 0. Pro ducts. Now we inv estigate pro ducts fro m the p oint o f view of to polo gical Hausdorﬀ dimension. By the pro duct o f t wo metric s pa ces we will alwa ys mean the l 2 -pro duct, that is, d X × Y (( x 1 , y 1 ) , ( x 2 , y 2 )) = q d 2 X ( x 1 , x 2 ) + d 2 Y ( y 1 , y 2 ) . First we r e call a well-kno wn statement, see [7, Chapter 3] and [7, Pro duct for - m ula 7.3] for the deﬁnition of the upper b ox-coun ting dimension and the pro of, resp ectively . In fact, [7] works in Euclidean spa c es only , but the pr o of g o es thr o ugh verbatim to g e neral metric spaces . Lemma 4.20 . L et X , Y b e non-empty metric sp ac es such that dim H Y = dim B Y , wher e dim B is the upp er b ox-c ounting dimension. Then dim H ( X × Y ) ≤ dim H X + dim H Y . Now we prove our next theorem which provides a large class of sets for w hich the to p olo gical Hausdor ﬀ dimension a nd the Hausdorﬀ dimension coincide. A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 13 Theorem 4.2 1. L et X b e a non-empty sep ar able metric sp ac e. Then dim tH ( X × [0 , 1]) = dim H ( X × [0 , 1 ]) = dim H X + 1 . Pr o of. F rom Theorem 4.4 it follows that dim tH ( X × [0 , 1]) ≤ dim H ( X × [0 , 1]). Applying Lemma 4.2 0 for Y = [0 , 1 ] we deduce that dim H ( X × [0 , 1]) ≤ dim H X + dim H [0 , 1 ] = dim H X + 1 . Finally , we pr ov e that dim H X + 1 ≤ dim tH ( X × [0 , 1]). Let us deﬁne pr X : X × [0 , 1 ] → X as pr X ( x, y ) = x a nd let Z = X × [0 , 1]. Theore m 3 .6 implies tha t there is a s e t A ⊆ Z s uch that dim H A ≤ dim tH Z − 1 and dim t ( Z \ A ) ≤ 0 . Since Z \ A is totally disco nnected, A intersects { x } × [0 , 1] for all x ∈ X , thus pr X ( A ) = X . Pro jections do not increase the Hausdorﬀ dimensio n, thus dim tH Z − 1 ≥ dim H A ≥ dim H pr X ( A ) = dim H X . Hence dim tH ( X × [0 , 1]) ≥ dim H X + 1, and the pr o of is complete.  Remark 4. 22. W e cannot drop sepa rability here. Indeed, if X is a n unco untable discrete metr ic space then it is not diﬃcult to see that dim tH ( X × [0 , 1]) = 1 and dim H ( X × [0 , 1]) = dim H X = ∞ . Separability is a ra ther natural assumption throug hout the pap er. First, the Hausdorﬀ dimensio n is only meaningful in this co ntext (it is alwa y s inﬁnite for non-separa ble spaces ), seco ndly for the theory o f top ological dimensio n this is the most usual framework. Corollary 4.23 . If X is a non-empty sep ar able metric sp ac e then dim tH  X × [0 , 1] d  = dim H  X × [0 , 1] d  = dim H X + d. The p ossibl e v alues of (dim t X , dim tH X , dim H X ) . No w we provide a com- plete des cription of the p ossible v alues o f the triple (dim t X , dim tH X , dim H X ). Moreov er, all p ossible v a lue s can b e r ealized by compact spa ces as well. Theorem 4.2 4. F or a triple ( d, s, t ) ∈ [0 , ∞ ] 3 the fol lowing ar e e quivalent. (i) Ther e exists a c omp act metric sp ac e K such that dim t K = d , dim tH K = s , and dim H K = t . (ii) Ther e exists a sep ar able metric sp ac e X su ch that dim t X = d , dim tH X = s , and dim H X = t . (iii) Ther e exists a metric sp ac e X such that dim t X = d , dim tH X = s , and dim H X = t . (iv) d = s = t = − 1 , or d = s = 0 , t ∈ [0 , ∞ ] , or d ∈ N + ∪ {∞} , s, t ∈ [1 , ∞ ] , d ≤ s ≤ t . Pr o of. T he implications ( i ) = ⇒ ( ii ) and ( ii ) = ⇒ ( iii ) are o b vious, and ( iii ) = ⇒ ( iv ) can ea sily b e chec k ed using F act 4.1 and Theo rem 4.4. It remains to prov e that ( iv ) = ⇒ ( i ). First, the empty set ta k es ca re of the cas e d = s = t = − 1. Let now d = s = 0, t ∈ [0 , ∞ ]. F o r t ∈ [0 , ∞ ] let K t be a Cantor set with dim H K t = t . Such sets are well-known to exist alr eady in [0 , 1] n for la rge enough n in ca se t < ∞ , whereas if C is the middle-thirds Cantor set then C N is such a set for t = ∞ . Then clearly dim t K t = dim tH K t = 0 and dim H K t = t , so we a re done with this case. Finally , let d ∈ N + ∪ {∞} , s, t ∈ [1 , ∞ ] , d ≤ s ≤ t . W e may as sume d < ∞ , o th- erwise the Hilb ert cub e provides a suitable ex ample. (Indeed, clear ly dim t [0 , 1 ] N = 14 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S dim tH [0 , 1 ] N = dim H [0 , 1 ] N = ∞ .) Deﬁne K d,s,t = ( K s − d × [0 , 1] d ) ∪ K t (this can b e understo o d a s the disjoint sum of metric spaces, but we may also ass ume that a ll these spac es are in the Hilb ert cub e, so the union is well deﬁned). Since dim t ( X × Y ) ≤ dim t X + dim t Y for non-empty spaces (see e.g. [6]), we obtain dim t ( K s − d × [0 , 1] d ) = 0 + d = d . Hence, by the stability o f the top ologica l dimension for clo sed sets, dim t K d,s,t = max  dim t  K s − d × [0 , 1 ] d  , dim t K t  = max { d, 0 } = d . Using Cor o llary 4.23 and the stability of the top olog ical Hausdor ﬀ dimension for closed sets we infer that dim tH K d,s,t = max  dim tH  K s − d × [0 , 1 ] d  , dim tH K t  = max { s − d + d, 0 } = s . Again by Coro llary 4.23 and b y the stability of the Hausdorﬀ dimension we obtain that dim H K d,s,t = max  dim H  K s − d × [0 , 1 ] d  , dim H K t  = max { s − d + d, t } = max { s, t } = t . This completes the pro o f.  The top ological Hausdo rﬀ dimens ion i s not a functio n o f the top olog i- cal and the Hausdorﬀ dimensio n. As a pa rticular case o f the ab ov e theor em we obtain that there a re compa c t metric spa ces X and Y such that dim t X = dim t Y and dim H X = dim H Y but dim tH X 6 = dim tH Y . This immediately implies the fol- lowing, which shows that the topo logical Hausdo r ﬀ dimension is indee d a genuinely new concept. Corollary 4. 25. dim tH X c annot b e c alculate d fr om dim t X and dim H X , even for c omp act metric sp ac es. 5. Calcula ting the topol ogical Hausdorff dimension 5.1. So m e classical fractals. First we present cer ta in natural exa mples of co m- pact sets K with dim t K = dim tH K < dim H K . Let S b e the Sier pi´ nski tria ngle, then it is well-kno wn that dim t S = 1 and dim H S = log 3 log 2 . Theorem 5.1 . L et S b e the Sierpi´ nski triangle. Then dim tH ( S ) = 1 . Pr o of. L e t ϕ i : R 2 → R 2 ( i = 1 , 2 , 3 ) b e the three similitudes with ratio 1 / 2 for which S = S 3 i =1 ϕ i ( S ). Sets of the form ϕ i n ◦ · · · ◦ ϕ i 1 ( S ), n ∈ N , j ∈ { 1 , . . . , n } , i j ∈ { 1 , 2 , 3 } ar e called the elementary pieces of S . It is not diﬃcult to see that U = { int S H : H is a ﬁnite union of elementary pieces o f S } is a basis of S such tha t # ∂ S U is ﬁnite fo r every U ∈ U . Therefor e dim H ∂ S U ≤ 0 , and hence dim tH S ≤ 1. On the other hand, S contains a line segment, ther e fore dim tH S ≥ dim tH [0 , 1 ] = 1 by mo notonicity .  Now we turn to the von K o c h snowﬂak e curve D . Recall that dim t D = 1 and dim H D = log 4 log 3 . F act 5 .2. If K is home omorphic to [0 , 1] then dim tH K = 1 . Pr o of. B y Coro llary 4.3 we o bta in that dim tH K ≥ 1. O n the other hand, since K is homeomo rphic to [0 , 1], there is a basis in K such that # ∂ U ≤ 2 for every U ∈ U . Thu s dim tH K ≤ 1.  Corollary 5.3. L et D b e the von Ko ch curve. Then dim tH D = 1 . Next we ta ke up a natural example o f a co mpact set K with dim t K < dim tH K < dim H K . Let T b e the Sierpi ´ nski carp et, then it is well-known that dim t T = 1 and dim H T = log 8 log 3 . A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 15 Theorem 5.4 . L et T b e the Sierpi´ nski c arp et . Then dim tH T = log 2 log 3 + 1 = log 6 log 3 . Pr o of. L e t C denote the middle-thirds Ca n tor set. Observ e that C × [0 , 1] ⊆ T . Then mo notonicity and Theorem 4.21 yield dim tH T ≥ dim tH ( C × [0 , 1]) = dim H C + 1 = log 2 log 3 + 1. Let us now prov e the opp osite inequality . F or n ∈ N and i = 1 , . . . , 3 n let z n i = 2 i − 1 2(3 n ) . Then clear ly { z n i : n ∈ N , i ∈ { 1 , . . . , 3 n }} is dens e in [0 , 1 ]. Let L b e a horizontal line deﬁned by an equation of the form y = z n i or a vertical line deﬁned by x = z n i . It is ea sy to see tha t L ∩ T consists of ﬁnitely many sets g eometrically similar to the middle-thirds Cantor s et. Using these lines it is not diﬃcult to construct a rectang ular basis U of T s uc h that dim H ∂ T U = log 2 log 3 for every U ∈ U , and hence dim tH T ≤ log 2 log 3 + 1.  Finally we re ma rk that, by Theorem 4 .21, K = C × [0 , 1] (where C is the middle- thirds Cantor set) is a natur a l example of a co mpact se t with dim t K < dim tH K = dim H K . 5.2. Kakey a s ets. Deﬁnition 5.5. A subset o f R d is called a Kakeya set if it contains a no n-degenerate line segment in every direction (some authors call thes e sets Besic ovitch sets ). According to a surpr ising c la ssical r esult, Kakey a sets of Leb esg ue measure zero exist. How ever, one of the most famo us co njectures in analysis is the Ka k eya Conjecture stating that every Kakey a se t in R d has Hausdo rﬀ dimension d . This is known to hold o nly in dimensio n at most 2 s o far, and a s olution alr eady in R 3 would have a huge impact on numerous a r eas of mathematics. It would b e tempting to attack the Kakey a Conjecture using dim tH K ≤ dim H K , but the following theorem, the main theor em of this subsec tio n will show that unfortunately we cannot get anything non-trivia l this way . Theorem 5. 6. Ther e exists a Kakeya set K ⊆ R d of top olo gic al Hausdorﬀ dimen- sion 1 for every inte ger d ≥ 1 . This result is of course sha r p, since if a set contains a line segment then its top ological Haus do rﬀ dimension is at lea st 1. W e will actually pr ov e so mewhat more, since we will es sent ially show that the generic ele men t of a ca refully chosen space is a Ka key a s e t of top ological Hausdo r ﬀ dimension 1. This idea, a s well as most of the others in this subsection are alr eady present in [17] by T. W. K¨ or ner. Howev e r , he only works in the pla ne and his s pa ce slightly diﬀers from ours. F or the sake of completeness we provide the rather short pro of in detail. Let ( K , d H ) b e the set o f non-empty compa ct subsets of R d − 1 × [0 , 1] endow ed with the Hausdo rﬀ metric, that is, for ea ch K 1 , K 2 ∈ K d H ( K 1 , K 2 ) = min { r : K 1 ⊆ B ( K 2 , r ) and K 2 ⊆ B ( K 1 , r ) } . It is well-known that ( K , d H ) is a complete metric space, see e.g. [1 4]. Let Γ = { ( x 1 , . . . , x d − 1 , 1) : 1 / 2 ≤ x i ≤ 1 , i = 1 , . . . , d − 1 } 16 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S denote a subset o f directions in R d . A clos e d line s e gment w connecting R d − 1 × { 0 } and R d − 1 × { 1 } is called a standard segment. Let us denote by F ⊆ K the system of those compact sets in R d − 1 × [0 , 1] in which for each v ∈ Γ w e can ﬁnd a standard s e gment w para llel to v . First we show that F is closed in K . Let us assume that F n ∈ F , K ∈ K and F n → K with resp ect to d H . W e have to s how that K ∈ F . Let v ∈ Γ b e a rbitrary . Since F n ∈ F , there exists a w n ⊆ F n parallel to v for every n . It is easy to see that S n ∈ N F n is bo unded, hence we c a n cho ose a subsequence n k such that w n k is conv ergent with resp ect to d H . But then clearly w n k → w for some sta ndard seg men t w ⊆ K , and w is paralle l to v . Hence K ∈ F indeed. Therefore, ( F , d H ) is a complete metr ic space and hence we ca n use Baire cate- gory a rguments. The next lemma is bas e d on [17, Thm. 3 .6.]. Lemma 5.7. The generic set in ( F , d H ) is of top olo gic al Hausdorﬀ dimension 1. Pr o of. T he rational cub es form a basis of R d , and their bounda r ies are covered by the r ational hyperpla ne s or thogonal to one o f the usual basis vectors o f R d . Therefore, it suﬃces to show that if S is a ﬁxed hyper plane o rthogonal to one of the us ua l basis vectors then { F ∈ F : dim H ( F ∩ S ) = 0 } is co -meager. F or n ∈ N + deﬁne F n =  F ∈ F : H 1 n 1 n ( F ∩ S ) < 1 n  . In o rder to show that { F ∈ F : dim H ( F ∩ S ) = 0 } = T n ∈ N + F n is co-meag e r , it is enough to prove that each F n contains a dense op en set. F or p ∈ R d , v ∈ Γ and 0 < α < π / 2 w e denote by C ( p, v , α ) the following do ubly inﬁnite clo sed cone C ( p, v , α ) = { x ∈ R d : the angle be tw een the lines o f v and x − p is a t most α } . W e denote b y V ( C ( p, v, α )) the set of those vectors u = ( u 1 , ..., u d − 1 , 1) for which there is a line in int( C ( p, v , α )) ∪ { p } parallel to u . Then V ( C ( p, v , α )) is r elatively op en in R d − 1 × { 1 } . The se ts of the form C ′ ( p, v , α ) = C ( p, v , α ) ∩  R d − 1 × [0 , 1 ]  will b e called trun- cated c o nes, and the system of truncated cones will b e denoted by C ′ . A truncated cone C ′ ( p, v , α ) is S -compatible if either C ′ ( p, v , α ) ∩ S = { p } , or C ′ ( p, v , α ) ∩ S = ∅ . The se t o f S -compatible truncated cones is denoted by C ′ S . Deﬁne F S as the set of those F ∈ F tha t can be written as the union of ﬁnitely many S -compatible truncated co nes and ﬁnitely many points in R d − 1 × [0 , 1 ]. Next we chec k that F S is dense in F . Suppo se F ∈ F is arbitrary a nd ε > 0 is given. First choose ﬁnitely man y po in ts { y i } t i =1 in F suc h that F ⊆ B ( { y i } t i =1 , ε ). Let v ∈ Γ be arbitr a ry , then there exists a standard segment w v ⊆ F parallel to v . By the choice of S and Γ, clearly w v * S , hence we ca n choo se p v and α v such that C ′ ( p v , v , α v ) ∈ C ′ S and d H ( C ′ ( p v , v , α v ) , w v ) ≤ ε . Obviously v ∈ V ( C ( p v , v , α v )), so { V ( C ( p v , v , α v )) } v ∈ Γ is an op en cov er o f the compact set Γ. Ther efore, there ar e { C ′ ( p v i , v i , α v i ) } m i =1 in C ′ S such that Γ ⊆ S m i =1 V ( C ( p v i , v i , α v i )). Put F ′ = S m i =1 C ′ ( p v i , v i , α v i ) ∪ { y 1 , . . . , y t } , then F ′ ∈ F S . It is ea sy to see that S m i =1 C ′ ( p v i , v i , α v i ) ⊆ B ( F , ε ), and combining this with { y i } t i =1 ⊆ F we obtain that F ′ ⊆ B ( F , ε ). By the choice of { y i } t i =1 we also have F ⊆ B ( F ′ , ε ). Th us d H ( F, F ′ ) ≤ ε . A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 17 Now using our dense set F S we v erify that F n contains a dense o pen set U . W e construct for all F 0 ∈ F S a ball in F n centered at F 0 . By the deﬁnition o f S -compatibility F 0 ∩ S is ﬁnite. Hence we ca n easily choos e a relatively op en set U 0 ⊆ S s uc h that F 0 ∩ S ⊆ U 0 and H 1 n 1 n ( U 0 ) < 1 n . Let us de ﬁne U = { F ∈ F : F ∩ S ⊆ U 0 } . Clearly F 0 ∈ U , U ⊆ F n and it is e a sy to see that U is op en in F . This completes the pr o of.  F ro m this we obtain the ma in theorem of the subsection a s follows. Pr o of of The or em 5.6. By the ab ov e lemma we can choose F 0 ∈ F s uc h that dim tH F 0 = 1. Then F 0 contains a line segment in every dir ection of Γ, hence we can choose ﬁnitely many is ometric copies of it, { F i } n i =1 such that the co mpact s e t K = S n i =0 F i contains a line segment in every direction. By the Lipschitz inv aria nc e of the top olog ical Hausdorﬀ dimensio n dim tH F i = dim tH F 0 for all i , and by the stability o f the top ological Ha usdorﬀ dimension for clos ed sets dim tH K = 1.  5.3. Brownian motion. One of the most importa n t sto ch astic pro cesses is the Brownian motio n (see e.g. [25]). Its range and graph also serve as imp ortant examples of fractal sets in geometric meas ure theory . Since the graph is always homeomorphic to [0 , ∞ ), F act 5.2 a nd countable s tabilit y for closed s ets yield that its top olog ic al Hausdorﬀ dimens io n is 1. Hence we fo cus o n the ra nge only . Each statement in this paragr aph is to b e under sto o d to hold with pro bability 1 (almost surely). Clea rly , in dimensio n 1 the rang e is a no n- degenerate interv al, so it has top ologica l Hausdorﬀ dimension 1. Moreover, if the dimensio n is at least 4 then the ra nge ha s no multiple points ([25]), so it is homeomorphic to [0 , ∞ ), which in turn implies as ab ove that the r ange has top olog ical Haus dorﬀ dimension 1 ag ain. Problem 5 .8. L et d = 2 or 3 . Determine the almo st sur e top olo gic al Hausdorﬀ dimension of the r ange of the d -dimensiona l Br ownian motion. Equivalently, de- termine the smal lest c ≥ 0 such that the r ange c an b e de c omp ose d int o a total ly disc onne ct e d set and a set of Hau s dorﬀ dimension at most c − 1 almost sur ely. Indeed, the tw o formulations of the pro blem ar e equiv a len t by Theorem 3.7. The following op en pro blem of W. W er ner [25, p. 38 4.] is closely related to P roblem 5.8 in the case d = 2. Notation 5. 9. B y a cur ve we mean a contin uous map γ : [0 , 1] → R d . Let us deno te by Ran( γ ) the r ange of γ . If γ : [0 , 1] → R 2 is a closed curve and p ∈ R 2 \ Ran( γ ) then let us deno te by W N ( γ , p ) the winding numb er o f γ with resp ect to p , see [8] for the deﬁnition. Problem 5.10 (W. W erner) . Is it t rue almost sur ely for t he planar Br ownian motion B : [0 , ∞ ) → R 2 that for every x, y ∈ R 2 \ Ran( B ) ther e exists a curve γ : [0 , 1] → R 2 such t hat γ (0) = x , γ (1 ) = y , and Ran( γ ) ∩ Ran( B ) is ﬁ nite? Remark 5.11. An aﬃrma tive answer to Proble m 5 .10 would pr obably also solve Problem 5.8. More precis ely , if there exis ts a n y function b : (0 , ∞ ) → (0 , ∞ ) with lim x → 0+ b ( x ) = 0 such that the ab ov e curve γ can b e cons tructed in the disc U ( x, b ( | x − y | )) then one ca n build a basis o f Ran( B ) as follows. Let x ∈ Ran( B ) 18 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S and ε > 0 b e given, a nd pick δ > 0 such that | x − y | < δ implies b ( | x − y | ) < ε 2 . Select a sequence of p oints x 0 , x 1 , . . . , x n = x 0 on the circle of r adius ε 2 centered at x such tha t x i / ∈ Ran( B ) and | x i − x i +1 | < δ for every i = 0 , . . . , n − 1 . Mo reov er, we c an a ls o assume that the argument of the vectors x i − x ( i = 0 , . . . , n − 1) is an incr easing se quence in [0 , 2 π ). This implies that if for e very i = 0 , . . . , n − 1 we construct a curve γ i from x i to x i +1 in U ( x i , ε 2 ) in tersecting Ran( B ) ﬁnitely many times, a nd glue these c ur ves together to obtain a c losed curve γ then the winding num b er W N ( γ , x ) = 1, and hence x is in a b ounded c ompo nen t U x,ε of R 2 \ Ran( γ ) se e [8, Prop ositio n 3 .16.]. Then it is ea sy to see that ∂ U x,ε ⊆ Ran( γ ), hence ∂ U x,ε ∩ Ra n( B ) is ﬁnite, a nd x ∈ U x,ε ⊆ U ( x, ε ). Therefore , the sets of the form U x,ε ∩ Ran( B ) form a ba s is of Ran( B ) such that the b oundary o f U x,ε relative to Ran( B ) is ﬁnite for a ll x and ε , thus the top olog ical Hausdorﬀ dimension o f Ran( B ) is 1 a lmost surely . While we can solve ne ither Pr oblem 5.8 nor Pr oblem 5.10, we are able to solve their Ba ir e categ o ry duals. First we need s ome prepa r ation. Deﬁnition 5 .12. Let us denote by C d the space of curves from [0 , 1] to R d endow e d with the s uprem um metric. As this is a c o mplete metric spa c e , we can use Baire category ar gument s. A standar d line se gment in R d is a closed non-degene r ate line s egment pa rallel to one of the standard ba sis vectors e 1 = (1 , 0 , . . . , 0) , . . . , e d = (0 , 0 , . . . , 1). The following theo rem g iv es an aﬃrma tive answer to the category dua l o f Pr ob- lem 5.8. Theorem 5.13. The r ange of t he generic f ∈ C d has top olo gic al H au s dorﬀ dimen- sion 1 for every d ∈ N + . Pr o of. If d = 1 then the statement is s tr aightforw ard, thus we may as sume d > 1. By Corolla ry 4.3 the rang e of every no n-constant f ∈ C d has top ologica l Hausdo r ﬀ dimension at lea st 1, thus we need to show that the generic f ∈ C d has top ologica l Hausdorﬀ dimension a t most 1. The r ational cub es form a basis o f R d , and their bo undaries a r e covered by countably many h yp erplanes, therefor e it is eno ug h to show that if S is a ﬁx ed hyperplane then { f ∈ C d : dim H (Ran( f ) ∩ S ) = 0 } is co- meager. By rota ting our co o rdinate sy s tem we may supp ose that S is not pa rallel to any vector in the s tandard ba s is { e 1 , . . . , e d } . F o r n ∈ N + deﬁne F n =  f ∈ C d : H 1 n 1 n (Ran( f ) ∩ S ) < 1 n  . In o rder to show that { f ∈ C d : dim H (Ran( f ) ∩ S ) = 0 } = T n ∈ N + F n is co-mea ger, it is enoug h to pro ve that e a ch F n is a dense op en se t. Let us ﬁx n ∈ N + . The regular ity of H 1 n 1 n implies tha t F n is op en. Le t G be the set o f curves f ∈ C d such that Ran( f ) is a union of ﬁnitely man y standard line s egment s. It is easy to see that G is dense in C d . As Ran( f ) ∩ S is ﬁnite for e v ery f ∈ G , we have G ⊆ F n . Thu s F n is dense in C d , and the pro of is co mplete.  Deﬁnition 5. 1 4. Let C ⊆ R 2 and le t p, q ∈ R 2 \ C . W e say that C sep ar ates p and q if these p oints b elong to diﬀerent connected comp onents of R 2 \ C . A metric space is called a c ontinuum if it is compact and connected. A set V ⊆ R 2 is a standar d op en set if it is a union of ﬁnitely ma n y ax is-parallel o pen square s. A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 19 W e say that f ∈ C 2 is a standar d curve if there exist 0 = x 0 < · · · < x m = 1 such that f maps the interv al I k = [ x k − 1 , x k ] bijectively to a standard line segment, and the standard line segments f ( I k ) and f ( I k +1 ) are orthogo nal for all k ∈ { 1 , . . . , m } , where we us e the notation I m +1 = I 1 . Notice that the p o in ts x k are uniquely determined. W e say that f ( x k ) and I k are the turn ing p oints and e dge intervals of f , resp ectively . A simple closed standard cur v e γ ∈ C 2 is a st andar d p olygon . Let α > 0. Let us deﬁne w ∈ C 2 to b e an α - wir e if there exists a standar d po lygon γ ∈ C 2 with edg e interv als I 1 , . . . , I m such that the lengths o f γ ( I k ) are at most α , and if E k denotes the segment we obtain from the edge γ ( I k ) by expanding it by leng th α in both directions , then w ( I k ) = E k for all k ∈ { 1 , . . . , m } , see Figure 1. W e say that the E k are the e dges of the α - wire w , and E k and E k +1 are adjac ent if k ∈ { 1 , . . . , m } , where E m +1 = E 1 . O f co ur se, w pas ses thro ug h p oints of w ( I k ) \ γ ( I k ) mo r e than once. Ran( γ ) Ran( w ) z}|{ α Figure 1. A standard p olyg on γ and the corr esp o nding α -wir e w The ab ov e de ﬁnitio ns ea sily imply the following facts. F act 5 .15. The standar d curves form a dense set in C 2 . F act 5 . 16. L et E , E ′ ⊆ R 2 b e two adjac ent e dges of an α -wir e. Assume t hat f ∈ C 2 and I , I ′ ⊆ [0 , 1] ar e disjoint close d intervals su ch t hat f ( I ) = E and f ( I ′ ) = E ′ . Then for al l g ∈ U ( f , α/ 2) we obtain that g ( I ) ∩ g ( I ′ ) 6 = ∅ . The next theorem a nswers the categ ory dual of Pr oblem 5.10 in the ne g ative. Theorem 5. 17. F or the generic f ∈ C 2 if p, q ∈ R 2 \ Ran( f ) and Ran( f ) sep ar ates p and q then for every c ontinuum C ⊆ R 2 with p , q ∈ C t he interse ction C ∩ Ran( f ) has c ar dinality c ontinuum. Before proving Theorem 5 .17 we need s ome lemmas. Lemma 5. 18. L et f ∈ C 2 b e a standar d curve and let p, q ∈ R 2 \ Ra n( f ) . If V ⊆ R 2 is a standar d op en set su ch that V ∩ Ran( f ) sep ar ates p and q then ther e is a standar d p olygo n γ ∈ C 2 such t hat Ran( γ ) ⊆ V ∩ Ran( f ) sep ar ates p and q . Pr o of. W e say that an op en line segment S is a mar ginal se gment if S ⊆ V ∩ Ra n( f ) and one of its endpo in ts is in ∂ V and the other endpo in t is either in ∂ V , or is 20 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S a turning p oint, or is a p oint attained by f mo re than once . Let S 1 , . . . , S n ⊆ V ∩ Ran( f ) b e the marg ina l segments. Clearly , they a re pairwise dis jo in t. Set K = ( V ∩ Ran( f )) \ ( S n i =1 S i ). Then K ⊆ V ∩ Ran( f ) is compact. First we prove that K separates p and q . Ass ume to the con trary that this is not the case, then ther e is a sta ndard curve g ∈ C 2 with g (0) = p and g (1) = q such that K ∩ Ran( g ) = ∅ . Let y 1 be a n endpo int of S 1 in ∂ V . If g meets S 1 then we can mo dify g in a small neig h b orho o d of S 1 such that the mo diﬁed curve passes thr ough y 1 and av oids S 1 ∪ K . Contin uing this pro cedure we obtain a c ur ve e g ∈ C 2 with e g (0) = p and e g (1 ) = q such that ( V ∩ Ran( f )) ∩ Ran( e g ) = ∅ , but this contradicts the fact that V ∩ Ran( f ) sepa rates p and q . Now elementary considerations show that there is a standa r d polyg on γ ∈ C 2 such that Ra n( γ ) ⊆ K . F or the sake of co mpleteness we mention that [2 6, Thm. 14 .3.] implies that K contains a comp onent C separating p a nd q . Then C is a lo cally connected contin uum, s o [32, (2.41)] yields that there exis ts a s imple clo s ed cur ve γ ∈ C 2 such tha t Ra n( γ ) ⊆ C separa tes p and q , a nd after r eparametriza tion γ will be a standa r d po lygon.  Lemma 5 .19. L et p, q ∈ R 2 and ε > 0 . As sume that f ∈ C 2 is a standar d cu rve and V ⊆ R 2 is a st andar d op en set su ch that V ∩ Ran( f ) sep ar ates p and q . Then ther e exist a standar d curve f 0 ∈ C 2 , st andar d op en sets V 0 , V 1 ⊆ R 2 and δ > 0 such that (i) V 0 , V 1 ⊆ V and dist( V 0 , V 1 ) > 0 , (ii) V j ∩ Ran( g ) sep ar ates p and q if j ∈ { 0 , 1 } and g ∈ U ( f 0 , δ ) , (iii) U ( f 0 , δ ) ⊆ U ( f , ε ) , (iv) f 0 = f on [0 , 1] \ f − 1 ( V ) . Pr o of. It is e no ugh to co ns truct a not necessar ily standard f 0 ∈ C 2 with the ab ov e prop erties, since by F a ct 5 .1 5 we can repla ce it with a sta ndard one. By Lemma 5.18 there exis ts a standard p olygon γ ∈ C 2 such that Ra n( γ ) ⊆ V ∩ Ran( f ) sepa rates p and q . Let Γ = Ran( γ ), w e may assume by decr easing ε if necessary that U (Γ , ε ) ⊆ V and p, q / ∈ U (Γ , ε ). Let us supp o se that q is in the b ounded comp onent of R 2 \ Γ. Then the winding num b ers W N ( γ , p ) = 0 and W N ( γ , q ) = ± 1, see [8, Pro pos ition 3 .16.] and [8, Pro pos ition 5.20 .]. It is easy to see that we can ﬁx a small enough α ∈ (0 , ε/ 9 ) such that there exist non-intersecting α -wires w 0 , w 1 ∈ U ( γ , ε/ 3), see Figure 2. Let W j = Ran( w j ) for j ∈ { 0 , 1 } , then clea rly W 0 ∩ W 1 = ∅ a nd W 0 ∪ W 1 ⊆ U (Γ , ε/ 3). Let us denote the edges of w 0 and w 1 by E 1 , . . . , E n 0 and E n 0 +1 , . . . , E n 0 + n 1 , r espec tively . Set n = n 0 + n 1 . Since W 0 ∪ W 1 ⊆ U (Γ , ε/ 3 ) ⊆ U (Ran( f ) , ε/ 3 ) and diam E k ≤ 3 α < ε/ 3, ther e exist distinct p oints z 1 , . . . , z n ∈ [0 , 1 ] such tha t E k ⊆ U ( f ( z k ) , 2 ε / 3) for all k ∈ { 1 , . . . , n } . By the contin uit y of f there are pairwise disjoint closed non-dege nerate interv als I 1 , . . . , I n ⊆ [0 , 1 ] such that E k ⊆ U ( f ( x ) , 2 ε/ 3) for all x ∈ I k and k ∈ { 1 , . . . , n } . Notice that each x ∈ S n k =1 I k satisﬁes f ( x ) ∈ U ( W 0 ∪ W 1 , 2 ε / 3) ⊆ U (Γ , ε ) ⊆ V , thus S n k =1 I k ⊆ f − 1 ( V ). Now for all k ∈ { 1 , . . . , n } deﬁne f 0 | I k such that f 0 ( I k ) = E k and let f 0 = f o n [0 , 1] \ f − 1 ( V ). It is easy to chec k that | f ( x ) − f 0 ( x ) | < 2 ε/ 3 for every x ∈ S n k =1 I k ∪  [0 , 1 ] \ f − 1 ( V )  . Then Tietze’s Extension Theo rem implies that we can extend f 0 contin uo us ly to [0 , 1] such that f 0 ∈ B ( f , 2 ε / 3). P rop erty (iv) follows from the deﬁnition of f 0 . A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 21 Γ Ran( w 0 ) Ran( w 1 ) Figure 2. Illustratio n to Lemma 5.19 Now let δ = min { α/ 2 , dist( W 0 , W 1 ) / 5 } > 0 . As f 0 ∈ B ( f , 2 ε/ 3 ) and δ ≤ α/ 2 < ε/ 3, we obtain U ( f 0 , δ ) ⊆ U ( f , ε ), thus pr op e rt y (iii ) ho lds . Let us pick standard op en sets V 0 and V 1 such that U ( W j , δ ) ⊆ V j ⊆ U ( W j , 2 δ ) for j ∈ { 0 , 1 } , then V j ⊆ U ( W j , 2 δ ) ⊆ U (Γ , 2 δ + ε/ 3) ⊆ U (Γ , ε ) ⊆ V , and dist( V 0 , V 1 ) ≥ dist( W 0 , W 1 ) − 4 δ ≥ δ > 0 , hence pro per t y (i) holds . Finally , let us ﬁx g ∈ U ( f 0 , δ ), we need to prove for prop erty (ii) that V 0 ∩ Ran( g ) separates p and q , and of cour se the same ar gument works for V 1 ∩ Ran( g ). Let us deﬁne the compact set K = S n 0 k =1 g ( I k ). Then S n 0 k =1 f 0 ( I k ) = W 0 yields K ⊆ U ( W 0 , δ ) ⊆ V 0 , thus K ⊆ V 0 ∩ Ran( g ). Therefore it is enough to prov e that K separates p and q . Consider 0 = x 0 < x 1 < · · · < x n 0 = 1 s uch that w 0 ([ x k − 1 , x k ]) = E k for ev ery k ∈ { 1 , . . . , n 0 } . Then g ∈ U ( f 0 , δ ) ⊆ U ( f 0 , α/ 2 ) and F act 5 .16 imply that g ( I k ) ∩ g ( I k +1 ) 6 = ∅ for a ll k ∈ { 1 , . . . , n 0 − 1 } and g ( I n 0 ) ∩ g ( I 1 ) 6 = ∅ . Thus we can c ho ose u k , v k ∈ I k for all k ∈ { 1 , . . . , n 0 } s uc h that g ( v k ) = g ( u k +1 ) and let h k : [ x k − 1 , x k ] → [ u k , v k ] b e a ho meomorphism with h k ( x k − 1 ) = u k and h k ( x k ) = v k , where we use the notations u n 0 +1 = u 1 and [ u, v ] = [min { u, v } , max { u, v } ]. Let us deﬁne ϕ : [0 , 1] → K such that ϕ | [ x k − 1 ,x k ] = g ◦ h k for a ll k ∈ { 1 , . . . , n 0 } . Then ϕ is a closed curve with Ra n( ϕ ) ⊆ K , so it is enoug h to prov e that Ran( ϕ ) sepa rates p a nd q . The function W N ( ϕ, · ) is constant on the comp onents of R 2 \ Ran( ϕ ) by [8, P r op osition 3.1 6.], thus it is s uﬃcien t to prove that W N ( ϕ, p ) 6 = W N ( ϕ, q ). F ro m g ∈ U ( f 0 , δ ), f 0 ( I k ) = E k and diam E k < ε/ 3 it follows that ϕ ∈ U ( w 0 , δ + ε/ 3 ) ⊆ U ( w 0 , 2 ε / 3). Thus w 0 ∈ U ( γ , ε/ 3 ) yields ϕ ∈ U ( γ , ε ). Hence ϕ a nd γ a re homoto pic in U (Γ , ε ) ⊆ R 2 \ { p, q } , so W N ( ϕ, p ) = W N ( γ , p ) = 0 and W N ( ϕ, q ) = W N ( γ , q ) = ± 1 , see [8, Corolla ry 3.8.]. The pr o o f is complete.  Lemma 5 .20. Assu me p, q ∈ R 2 and ε > 0 . L et f ∈ C 2 b e a standar d cu rve and let V 1 , . . . , V m ⊆ R 2 b e p airwise disjoint standar d op en sets such that V k ∩ Ran( f ) 22 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S sep ar ates p and q for al l k ∈ { 1 , . . . , m } . Then ther e exist a non-empty op en set V ⊆ U ( f , ε ) and standar d op en sets V kj ⊆ R 2 such that for al l k ∈ { 1 , . . . , m } and j ∈ { 0 , 1 } (i) V kj ⊆ V k and dist( V k 0 , V k 1 ) > 0 , (ii) V kj ∩ Ran( g ) sep ar ates p and q if g ∈ V . Pr o of. L e t g 0 = f and δ 0 = ε . Now for k ∈ { 0 , . . . , m − 1 } we construct by induction a standar d curve g k ∈ C 2 , a δ k > 0, and for all i ∈ { 1 , . . . , k } and j ∈ { 0 , 1 } sta ndard op en sets V ij such that for all i ∈ { 1 , . . . , k } and j ∈ { 0 , 1 } (1) V ij ⊆ V i and dist( V i 0 , V i 1 ) > 0 , (2) V ij ∩ Ran( g ) sepa rates p and q if g ∈ U ( g k , δ k ), (3) U ( g k , δ k ) ⊆ U ( f , ε ), (4) g k = f on [0 , 1] \ f − 1 ( V 1 ∪ · · · ∪ V k ). The case k = 0 is done, since (3) and (4) obviously hold, while (1 ) and (2) hold v acuously , since there are no s e ts V ij . Let us now take up the inductive step. Since g k = f on f − 1 ( V k +1 ) ⊆ [0 , 1] \ f − 1 ( V 1 ∪ · · · ∪ V k ) , we obtain that V k +1 ∩ Ran( g k ) separates p and q . Lemma 5.19 applied to the standard c ur ve g k , standard op en s et V k +1 and δ k > 0 yields that we can satisfy prop erties (1)-(4) for k + 1. Finally , the no n-empt y op en set V = U ( g m , δ m ) ⊆ U ( f , ε ) and the constructed standard op en sets V ij satisfy pr ope rties (i)-(ii).  Now we a re ready to pr ov e Theorem 5.17. Pr o of of The or em 5.17. Let F b e the set of functions f ∈ C 2 such that if Ra n( f ) separates a pair of p oints p a nd q then for every contin uum C ⊆ R 2 with p, q ∈ C the intersection C ∩ Ran( f ) has cardina lity co n tinuum . W e need to show that F is co-meag e r in C 2 . F or ﬁx e d p, q ∈ R 2 let F p,q be the set of functions f ∈ C 2 such that if Ran( f ) separ ates p and q then for every contin uum C ⊆ R 2 with p, q ∈ C the intersection C ∩ Ran( f ) has cardina lit y contin uum. As F = \ p,q ∈ Q × Q F p,q , it is enoug h to show that the sets F p,q are co-mea ger in C 2 . Let p, q ∈ R 2 , p 6 = q be a rbitrarily ﬁxed. In o rder to prov e that F p,q is co- meager, we play the Ba na ch - Mazur g ame in the metric space C 2 : First Play er I chooses a no n-empt y op en se t U 1 ⊆ C 2 , then Play er I I choo ses a no n-empt y op en set V 1 ⊆ U 1 , Player I contin ues with a non- empt y o pen set U 2 ⊆ V 1 , a nd so on. By deﬁnition P layer I I wins this game if T ∞ n =1 V n ⊆ F p,q . It is well-known that Player I I has a winning str ategy iﬀ F p,q is co-mea ger in C 2 , see [2 7, T hm. 1] or [14, (8.33 )]. Thus we need to prov e that Play er II has a winning strateg y . Now we describ e the strategy of Player I I. If there is an f ∈ U 1 such that p and q ar e in the same c o mpo ne nt of R 2 \ Ran( f ) then ther e is an ε > 0 such that all functions in U ( f , ε ) ⊆ U 1 hav e this pr ope r t y . Th us U ( f , ε ) ⊆ F p,q , and the strategy of P lay er I I is the following. Let V 1 = U ( f , ε ), and the other mov es of Play er I I are arbitrar y . Then clear ly T ∞ n =1 V n ⊆ F p,q , s o Player I I wins the g ame. If Ran( f ) sepa rates p a nd q fo r all f ∈ U 1 then the strategy of Play er I I is the following. The functions f with the pr o per t y p , q / ∈ Ran( f ) form a dense op en subset A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 23 in U 1 , s o by F act 5.1 5 there is a standard curve f 1 ∈ U 1 such that Ran( f 1 ) sepa rates p and q . Let V ⊆ R 2 be a s tandard op en set with Ran( f 1 ) ⊆ V . Let ε 1 > 0 b e so small that U ( f 1 , ε 1 ) ⊆ U 1 . Applying Lemma 5.20 for ε 1 , f 1 and V implies that there is a no n-empt y op en set V 1 ⊆ U ( f 1 , ε 1 ) a nd standar d op en sets V 0 , V 1 ⊆ V such that dist( V 0 , V 1 ) > 0 and V j ∩ Ran( g ) s eparates p a nd q for every j ∈ { 0 , 1 } and g ∈ V 1 . Next we supp ose tha t n ∈ N + and after the n th mov e o f Player I I the non-empty op en set V n and standard op en sets V j 1 ...j i ⊆ R 2 ( i ∈ { 1 , . . . , n } , j 1 , . . . , j i ∈ { 0 , 1 } ) are alr eady deﬁned such that for all j 1 , . . . , j n ∈ { 0 , 1 } (i) V j 1 ...j n ⊆ V j 1 ...j n − 1 and dist( V j 1 ...j n − 1 0 , V j 1 ...j n − 1 1 ) > 0 , (ii) V j 1 ...j n ∩ Ran( g ) sepa rates p and q for all g ∈ V n . Suppo se that Player I cont inues with U n +1 ⊆ V n . By F act 5.15 Play er I I ca n choose a s ta ndard curve f n +1 ∈ U n +1 . Then f n +1 ∈ V n implies that (ii ) holds for f n +1 . Let ε n +1 > 0 b e so small that U ( f n +1 , ε n +1 ) ⊆ U n +1 . Applying Lemma 5.2 0 for ε n +1 , f n +1 and the o pen sets V j 1 ...j n we obtain a non-empty op en set V n +1 ⊆ U ( f n +1 , ε n +1 ) ⊆ U n +1 and sta nda rd op en sets V j 1 ...j n +1 ⊆ R 2 witnessing that prop erties (i)-(ii) hold for n + 1. Finally , we need to pr ov e that Player I I wins the game with the ab ov e strateg y if Ran( f ) s eparates p and q for all f ∈ U 1 . L e t f ∈ T ∞ n =1 V n , w e need to prove that f ∈ F p,q . Let C ⊆ R 2 be a contin uum with p, q ∈ C , w e need to show that C ∩ Ran( f ) has cardinality co n tinu um. Pr o per t y (ii) implies that we can choose c j 1 ...j n ∈ C ∩ ( V j 1 ...j n ∩ Ran( f )) for every n ∈ N + and j 1 , . . . , j n ∈ { 0 , 1 } . If j = ( j 1 , j 2 , . . . ) ∈ { 0 , 1 } N then let c j be an arbitrary limit p oint of the set { c j 1 ...j n : n ∈ N + } . Then the c o mpactness of C and Ran( f ) implies that c j ∈ C ∩ Ran( f ) for every j ∈ { 0 , 1 } N . Th us it is enough to prov e that if j , k ∈ { 0 , 1 } N , j 6 = k then c j 6 = c k . Assume that n ∈ N + is the minimal num b er s uch that the n th co ordinate of j and k diﬀer. As the sets V j 1 ...j i form a nested sequence by prop erty (i ), we o btain c j ∈ cl V j 1 ...j n and c k ∈ cl V k 1 ...k n , a nd proper ty (i) also yields cl V j 1 ...j n ∩ cl V k 1 ...k n = ∅ . Therefore c j 6 = c k , a nd the pro of is c omplete.  6. Applica tion I: Mand elbr ot’s fract al percola tion process In this s e ction we take up one of the most imp ortant rando m fractals, the limit set M ⊆ R 2 of the fr a ctal p ercola tion pro cess deﬁned by Mandelbro t in [19]. His orig inal motiv a tio n was that this mo del ca ptures certain fea tures of turbu- lence, but then this rando m set turned out to b e very interesting in its own right. F or ex a mple, M serves a s a very powerful to ol for calculating Ha us dorﬀ dimens io n. It is w ell known that for each γ ∈ (0 , 2) there is a fracta l p e r colation M ( γ ) such that dim H M ( γ ) = 2 − γ almos t surely , pr ovided that M ( γ ) 6 = ∅ . Hawk e s [10] ha s shown that for a ﬁxed clos e d s et C in the unit s quare M ( γ ) ∩ C = ∅ almo st surely if dim H C < γ , and M ( γ ) ∩ C 6 = ∅ with p ositive probability if dim H C > γ . There- fore { M ( γ ) : γ ∈ (0 , 2) } ser ves as a family of r andom test sets : If we know which per colation limit s ets M ( γ ) intersect C with p ositive pr obability , we can determine dim H C . The idea of using r andom test sets go es ba ck to T aylor [30], while the use of p ercola tio n limit s ets as test s ets is due to Kho shnevisan, Peres, and Xiao [1 6]. In the context of trees, similar ideas were develop ed by Lyons [18]. Moreov er, it can b e shown that the ra nge of a d -dimensiona l Brownian motion is interse ction-e quivalent to a fractal p ercola tion, tha t is, they intersect all clo sed subsets o f the d -dimensional unit cub e with comparable pr obabilities. This c an b e 24 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S used to deduce numerous dimension r e lated re sults ab out B rownian motion. F or more on these results see Peres [28, 29], see also [25]. Let us now formally describ e the fractal p ercola tion pro cess. Le t p ∈ (0 , 1 ) and n ≥ 2, n ∈ N b e ﬁxed. Set M 0 = M ( p,n ) 0 = [0 , 1] 2 . W e divide the unit square into n 2 equal closed sub-squa res of side-leng th 1 /n in the natura l wa y . W e keep ea c h sub-square indep endently with proba bility p (and er ase it with proba bility 1 − p ), and denote by M 1 = M ( p,n ) 1 the union o f the kept s ub-squares. Then each square in M 1 is divided into n 2 squares of side-length 1 /n 2 , a nd w e keep eac h of them independently (and als o indep endently o f the earlier c hoices) with proba bilit y p , etc. After k steps let M k = M ( p,n ) k be the union of the kept k th level square s with side-length 1 /n k . Let (6.1) M = M ( p,n ) = ∞ \ k =1 M k . The pro ces s we ha ve just describ ed is called Mandelbr ot’s fr actal p er c olation pr o c ess , and M is ca lled its limit set . Percolation frac tals are not only interesting from the p oint of view of turbulence and fractal geometry , but they ar e also closely related to the (usual, graph-theor etic) per colation theor y . In case of the fractal pe rcolation the ro le of the clusters is played by the connected comp onents. Our s tarting p oint will b e the following celebr ated theorem. Theorem 6. 1 (Chay es-Chay es-Durrett, [5]) . Ther e exists a critic al pr ob ability p c = p ( n ) c ∈ (0 , 1) su ch that if p p c then M c ontains a nontrivial c onne cte d c omp onent with p ositive pr ob ability. They a ctually pr ov e more, the most p ow erful version states that in the sup er- critical ca se (i.e. when p > p c ) there is actually a unique unbounded c ompo nen t if the pro cess is e x tended to the who le plane, but we will only concentrate on the most sur prising fact that the critical probability is strictly b etw een 0 and 1. The ma in go al of the present section will b e to prove the following genera liz a tion of the ab ov e theorem. Theorem 6. 2. F or every d ∈ [0 , 2) t her e exists a critic al pr ob ability p ( d ) c = p ( d,n ) c ∈ (0 , 1 ) such that if p p ( d ) c then dim tH M > d almost sur ely (pr ovide d M 6 = ∅ ). In or der to see that we actually obtain a gene r alization, just note that a com- pact space is totally disc onnected iﬀ dim t M = 0 ([6]), also that dim tH M = 0 iﬀ dim t M = 0 , and use d = 0. Theorem 6.1 bas ic ally says that certain curves show up at the critica l probability , and our pro of will show that even ‘thick’ families of curves s how up, where the word thic k is re la ted to larg e Hausdorﬀ dimens io n. In the rest of this s ection ﬁr st we do some preparatio ns in the ﬁr st subse c tion, then we prov e the main theorem (Theorem 6.2) in the next subsection, and ﬁnally give an upp er b ound for dim tH M and co nc lude that dim tH M < dim H M almost surely in the non-trivia l cases . 6.1. Preparation. F or the pro ofs of the statements in the next tw o r emarks see e.g. [5]. A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 25 Remark 6.3. It is w ell-known from the theory of branching pro cesses that M = ∅ almost s urely iﬀ p ≤ 1 n 2 , s o we may assume in the following that p > 1 n 2 . If 1 n 2 < p ≤ 1 √ n then dim t M = 0 a lmo st surely . Hence F act 4 .1 implies that dim tH M = 0 almost surely . (In fact, the sa me holds even for p 1 n 2 we hav e dim H M = 2 + log p log n almost s urely , provided M 6 = ∅ . W e will also need the 1-dimensional analogue of the pr o cess (interv als instead o f squares). Here M (1 D ) = ∅ almo st surely iﬀ p ≤ 1 n , and for p > 1 n we have dim H M (1 D ) = 1 + log p log n almost s urely , provided M (1 D ) 6 = ∅ . Now we chec k that the a lmo st sure top ologic a l Haus dorﬀ dimensio n of M als o exists. Lemma 6.5. F or every p > 1 n 2 and n ≥ 2 , n ∈ N ther e exists a numb er d = d ( p,n ) ∈ [0 , 2] su ch that dim tH M = d almost su r ely, pr ovide d M 6 = ∅ . Pr o of. L e t N b e the r a ndom num ber of squar e s in M 1 . Let us s e t q = P ( M = ∅ ). Then q < 1 by Remark 6.3, a nd [7, Thm. 15.2] gives that q is the least p ositive ro ot of the p olynomial f ( t ) = − t + n 2 X k =0 P ( N = k ) t k . Let us ﬁx an ar bitrary x ∈ [0 , ∞ ). As q > 0 , we obtain P (dim tH M ≤ x ) > 0. Firs t we show that P (dim tH M ≤ x ) is a r o ot of f . If N > 0 then le t M 1 = { Q 1 , . . . , Q N } , where the Q i are the ﬁrst level sub- squares. F o r e very i and k let M Q i k be the union of those squar es in M k that a re in Q i , a nd let M Q i = T k M Q i k . (Note that this is not the same a s M ∩ Q i , since in this latter set there may b e p oints on the b oundar y of Q i ‘coming fr o m squar es outside of Q i ’.) Then M Q i has the same dis tr ibution a s a similar copy of M (this is called statistical self-similar it y), and hence for every i P  dim tH M Q i ≤ x  = P (dim tH M ≤ x ) . Using the stability of the top olog ic al Hausdo rﬀ dimension for closed sets and the fact that the M Q i are indep e ndent and hav e the same dis tr ibution under the condition N = k > 0, this implies P (dim tH M ≤ x | N = k ) = P  dim tH M Q i ≤ x for each 1 ≤ i ≤ k | N = k  =  P  dim tH M Q 1 ≤ x   k =  P (dim tH M ≤ x )  k . 26 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S F or N = 0 we hav e P (dim tH M ≤ x | N = 0) = 1 , ther efore we obtain P (dim tH M ≤ x ) = n 2 X k =0 P ( N = k ) P (dim tH M ≤ x | N = k ) = n 2 X k =0 P ( N = k )  P (dim tH M ≤ x )  k , and thus P (dim tH M ≤ x ) is indeed a r o ot of f for every x ∈ [0 , ∞ ). As mentioned ab ov e, q 6 = 1 and q is also a ro ot of f . Mor eov e r , 1 is obviously also a ro ot, and it is easy to see that f is strictly co nvex on [0 , ∞ ), hence there are at most tw o nonnegative ro ots. Ther efore q a nd 1 are the only nonnega tiv e ro o ts, th us P (dim tH M ≤ x ) = q or 1 for every x ∈ [0 , ∞ ). If P (dim tH M ≤ 0) = 1 then we ar e done , so we may a ssume that P (dim tH M ≤ 0) = q . Then the distribution function F ( x ) = P (dim tH M ≤ x | M 6 = ∅ ) = P (dim tH M ≤ x ) − q 1 − q only a ttains the v alues 0 a nd 1 , a nd clea rly F (0) = 0 and F (2 ) = 1. Thus there is a v alue d where it ‘jumps’ from 0 to 1. This c oncludes the pro of.  6.2. Pro o f of Theorem 6 .2; the low er estimate o f dim tH M . Set p ( d,n ) c = sup n p : dim tH M ( p,n ) ≤ d a lmost surely o . First we ne e d so me lemmas. The following one is a nalogous to [9, p. 38 7]. Lemma 6.6. F or every d ∈ R and n ∈ N , n ≥ 2 p ( d,n ) c < 1 ⇐ ⇒ p ( d,n 2 ) c < 1 . Pr o of. C le a rly , it is eno ugh to show that (6.2) p ( d,n ) c  1 −  1 − p ( d,n ) c  1 n 2  ≤ p ( d,n 2 ) c ≤ p ( d,n ) c . W e say that the ra ndom construction X is dominated by the random co nstruction Y if they can b e rea lized o n the same probability spac e s uch that X ⊆ Y almos t surely . Let us ﬁrst prove the seco nd inequality in (6.2). It clearly suﬃces to show that dim tH M ( p,n 2 ) ≤ d a lmost surely = ⇒ dim tH M ( p,n ) ≤ d almost surely . But this is rather stra ight forward, since M ( p,n ) 2 k is easily se e n to b e dominated by M ( p,n 2 ) k for every k , hence M ( p,n ) is dominated by M ( p,n 2 ) . Let us now prove the ﬁrs t inequality in (6.2). Set ϕ ( x ) = 1 − ( 1 − x ) 1 /n 2 . W e need to show that (6.3) 0 < p < p ( d,n ) c ϕ ( p ( d,n ) c ) = ⇒ dim tH M ( p,n 2 ) ≤ d almost surely . Since xϕ ( x ) is a n increas ing homeomorphism o f the unit interv al, p = q ϕ ( q ) for some q ∈ (0 , 1 ). Then clearly q < p ( d,n ) c , so dim tH M ( q,n ) ≤ d almost surely . Therefore, in or der to prov e (6.3) it suﬃce s to chec k that (6.4) M ( p,n 2 ) is dominated by M ( q,n ) . A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 27 First we check that (6.5) M ( ϕ ( q ) ,n 2 ) k is dominated by M ( q,n ) k for every k , and conseque ntly M ( ϕ ( q ) ,n 2 ) is do minated by M ( q,n ) . Indeed, in the second ca se we erase a sub-square of side length 1 n with pr obability 1 − q a nd keep it with probability q , while in the ﬁrst c ase we c ompletely era se a sub- square of side length 1 n with the same proba bilit y (1 − ϕ ( q )) n 2 = 1 − q a nd hence keep at le ast a su bset of it with probability q . But this will easily imply (6.4), which will complete the pr o o f. Indeed, after ea ch step o f the pro cesses M ( ϕ ( q ) ,n 2 ) and M ( q,n ) let us per form the following pro ce dur es. F or M ( ϕ ( q ) ,n 2 ) let us keep every existing squar e indep endently with pr obability q and eras e it with pro babilit y 1 − q (we do no t do a n y sub divisions in this ca se). F or M ( q,n ) let us take one more step o f the construction o f M ( q,n ) . Using (6 .5) this easily implies that M ( qϕ ( q ) ,n 2 ) k is dominated by M ( q,n ) 2 k for every k , henc e M ( qϕ ( q ) ,n 2 ) is dominated by M ( q,n ) , but q ϕ ( q ) = p , and hence (6.4) ho lds.  F ro m now on let N b e a ﬁxed (large) p ositive integer to b e chosen later . Rec a ll that a squa re of level k is a set of the fo rm  i N k , i +1 N k  ×  j N k , j +1 N k  ⊆ [0 , 1] 2 . Deﬁnition 6.7. A walk of level k is a seq uence ( S 1 , . . . , S l ) of no n-ov e rlapping squares of lev el k such that S r and S r +1 are abutting for every r = 1 , . . . , l − 1, moreov er S 1 ∩ ( { 0 } × [0 , 1]) 6 = ∅ and S l ∩ ( { 1 } × [0 , 1]) 6 = ∅ . In pa rticular, the only walk of level 0 is ([0 , 1] 2 ). Deﬁnition 6.8. W e say that ( S 1 , . . . , S l ) is a turning walk (of level 1 ) if it satisﬁes the prop erties o f a walk of level 1 ex cept that instead o f S l ∩ ( { 1 } × [0 , 1]) 6 = ∅ we require that S l ∩ ([0 , 1 ] × { 1 } ) 6 = ∅ . Lemma 6.9. L et S b e a set of N − 2 distinct squar es of level 1 interse cting { 0 } × [0 , 1 ] , and let T b e a set of N − 2 distinct squar es of level 1 interse cting { 1 } × [0 , 1] . Mor e over, let F ∗ b e a squar e of level 1 such t hat the r ow of F ∗ do es not interse ct S ∪ T . Then ther e exist N − 2 non-overlapping walks of level 1 not c ontaining F ∗ such t hat the s et of their ﬁrst squar es c oincides with S and the set of t heir last squar es c oincides with T . (Se e Figur e 3.) Case 1. Case 2 . Case 3. Figure 3. Illustratio n to Le mma 6.9 28 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S Pr o of. T he pro of is by induction o n N . The case N = 2 is obvious. Case 1. F ∗ is in the top or b ottom r ow. By simply ignoring this row it is straightforward how to constr uct the walks in the r emaining rows. Case 2 . F ∗ is not in t he top or b ottom r ow, and b oth top c orners or b oth b ottom c orners ar e in S ∪ T . Without lo ss of genera lit y w e may supp ose that bo th top co rners are in S ∪ T . Let the straight w alk connecting these tw o cor ners be one of the walks to b e constr uc ted. Then let us shift the r emaining member s of T to the left by one sq uare, and either we can apply the induction hypothesis to the ( N − 1) × ( N − 1) many square s in the b ottom left corner of the original N × N man y squa res, or F ∗ is not among these ( N − 1 ) × ( N − 1 ) man y squar es and then the argument is even easier. Then one ca n see how to g et the required walks. Case 3. Neither Case 1 nor Case 2 holds. Since there are only tw o squar es missing on bo th sides, and F ∗ cannot b e the top or b ottom row, we infer that b oth S a nd T contain a t lea s t one corner . Since Case 2 do es not hold, we obtain that b oth the top left and the b ottom r ight cor ners or b oth the b ottom left a nd the top rig ht corners are in S ∪ T . Without los s of generality we may suppo se that b oth the top left and the bo ttom right corners are in S ∪ T . By reﬂec ting the picture ab out the center o f the unit squa re if neces s ary , we may as s ume that F ∗ is not in the rightmost column. W e now construct the ﬁrst walk. Let it r un straig h t from the top left cor ner to the top rig h t corner, a nd then contin ue down w ards unt il it ﬁr st re a ches a member o f T . The n, as ab ov e, we ca n similarly apply the inductio n hypo thesis to the ( N − 1) × ( N − 1) many squares in the b ottom left corne r , and we ar e done.  Lemma 6.10 . L et S b e a set of N − 2 distinct squar es of level 1 int erse cting { 0 } × [0 , 1] , and let T b e a set of N − 2 distinct squar es of level 1 interse cting [0 , 1 ] × { 1 } (t he sets of starting and t erm inal squar es). Mor e over, let F ∗ b e a squar e of level 1 (the forbidden squar e) su ch that the r ow of F ∗ do es not int erse ct S and the c olumn of F ∗ do es not interse ct T . Then ther e exist N − 2 non-overlapping turning walks not c ontaining F ∗ such that the set of t heir ﬁrst squar es c oincides with S and the set of their last squar es c oincides with T . Pr o of. O b vious, just take the simplest ‘L-shap ed’ walks.  The last tw o lemmas will almost immediately imply the following. Lemma 6.11 . L et ( S 1 , . . . , S l ) b e a walk of level k , and F a system of squar es of level k + 1 such that e ach S r c ontains at most 1 memb er of F . Then ( S 1 , . . . , S l ) c ontains N − 2 non-overlapping sub-walks of level k + 1 avoiding F . (Se e Figur e 4.) Pr o of. W e may assume that each S r contains e x actly 1 member of F . Let us denote the member of F in S r by F ∗ r . The s ub- w alks will b e co nstructed s eparately in ea c h S r , using an a ppropriately r o tated or reﬂected version of e ither Lemma 6.10 or L e mma 6.9. It suﬃces to c onstruct S r and T r for every r (compatible with F ∗ r ) so that for every member of T r there is an a butting member of S r +1 . (Of cours e we also have to make sure that every member of S 1 int ersects { 0 } × [0 , 1] and every mem b er of T l int ersects { 1 } × [0 , 1].) F or example, the cons tr uction of T r for r < l is as fo llows. The squares S r and S r +1 share a common edge E . Assume for simplicity that E is ho rizontal. Then T r will c onsist of those sub-squa r es of S r of level k + 1 A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 29 Figure 4. Illustratio n to Lemma 6.11 that intersect E a nd whos e co lumn diﬀers fr o m that of F ∗ r and F ∗ r +1 . If these tw o columns happ en to co incide then w e can arbitrarily e rase one more square . The remaining cons tructions ar e similar and the de ta ils are left to the re a der.  Deﬁnition 6. 12. W e say tha t a square in M k is 1 -ful l if it contains at least N 2 − 1 many sub- squares from M k +1 . W e say tha t it is m -ful l , if it contains at least N 2 − 1 many m − 1-full sub-s quares from M k +1 . W e call M ful l if M 0 is m -full for every m ∈ N + . The following lemma was the key realizatio n in [5 ]. Lemma 6.1 3. Ther e is a p ( N ) < 1 so that P  M ( p,N ) is ful l  > 0 for al l p > p ( N ) . See [5] or [7, Pro p. 15.5] for the pro of. Deﬁnition 6.1 4. Let L ≤ N b e p ositive int egers. A compact set K ⊆ [0 , 1] is called ( L, N )-r egular if it is of the form K = T i ∈ N K i , where K 0 = [0 , 1], and K k +1 is obtained by dividing every in terv al I in K k int o N man y non-ov erlapping clos ed int erv a ls of le ngth 1 / N k +1 , and choo sing L ma n y of them for each I . The following fact is well-known, see e.g. the mor e general [7, Thm. 9 .3]. F act 6 .15. An ( L, N ) -r e gular c omp act set has Hausdorﬀ dimension log L log N . Next we prove the main result of the present subsection. Pr o of of The or em 6.2. Let d ∈ [0 , 2) be arbitrary . Fir st we verify that, for suﬃ- ciently large N , if M = M ( p,N ) is full then dim tH M > d . The stra tegy is as follows. W e deﬁne a collection G of disjoint connected subsets of M such tha t if a set in- tersects each member o f G then its Hausdorﬀ dimension is lar g er than d − 1. The n we show tha t for every countable op en basis U o f M the union of the b oundaries, S U ∈U ∂ M U intersects each mem b er of G , which clear ly implies dim tH M > d . 30 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S Let us ﬁx an integer N such that (6.6) N ≥ 6 and log( N − 2) log N > d − 1 , and let us assume that M is full. Using Lemma 6 .13 at each step we can choo se N − 2 non-ov erlapping walks of level 1 in M 1 , then N − 2 no n-ov erlapping walks of level 2 in M 2 in each of the ab ov e walks, etc. Let us denote the obta ined system at step k by G k =  Γ i 1 ,...,i k : ( i 1 , . . . , i k ) ∈ { 1 , . . . , N − 2 } k  , where Γ i 1 ,...,i k is the union of the squa res of the co rresp onding walk. (Set G 0 = { Γ ∅ } = { [0 , 1] 2 } .) Let us also put C k = n y ∈ [0 , 1] : (0 , y ) ∈ [ G k o and deﬁne C = \ k ∈ N C k . Then cle a rly C is a n ( N − 2 , N )-reg ular compact set, ther efore F act 6.15 yields that dim H C = log( N − 2) log N > d − 1. As C \ Q 6 = ∅ , we hav e dim H ( C \ Q ) = dim H C > d − 1. F or every y ∈ C \ Q and ev ery k ∈ N there is a unique ( i 1 , . . . , i k ) suc h that (0 , y ) ∈ Γ i 1 ,...,i k . (F or a y of the form i N l there may be t wo s uc h ( i 1 , . . . , i k ), and we would like to avoid this c omplication.) P ut Γ k ( y ) = Γ i 1 ,...,i k and Γ( y ) = T ∞ k =1 Γ k ( y ). Since Γ( y ) is a decreas ing intersection of compact connected sets, it is itself co nnected ([6]). (Actually , it is a co n tin uous curve, but we will not nee d this here.) It is also ea sy to see that it intersects { 0 } × [0 , 1] a nd { 1 } × [0 , 1]. W e can now deﬁne G = { Γ( y ) : y ∈ C \ Q } . Next w e prov e that G co nsists of disjo in t sets. Let y , y ′ ∈ C \ Q b e distinct. Pick l ∈ N so larg e such tha t | y − y ′ | > 6 N l . Then there are a t lea s t 5 interv als of level l b etw ee n y and y ′ . Since we a lw ays chose N − 2 interv als out of N a long the construction, ther e ca n b e at most 4 consecutive non-selected interv als, there fore there is a Γ i 1 ,...,i l separating y and y ′ . But then this a lso separates Γ l ( y ) a nd Γ l ( y ′ ), hence Γ( y ) and Γ( y ′ ) a re disjoint. Now we chec k that for ev ery y ∈ C \ Q and every coun table op en ba s is U of M the se t S U ∈U ∂ M U intersects Γ( y ). Let z 0 ∈ Γ( y ) and U 0 ∈ U b e such that z 0 ∈ U 0 and Γ( y ) * U 0 . Then ∂ M U 0 m ust intersect Γ( y ), since otherwise Γ( y ) = (Γ( y ) ∩ U 0 ) ∪ (Γ( y ) ∩ int M ( M \ U 0 )), hence a connected set would be the union of tw o non-e mpty disjoint relatively op en sets, a contradiction. Thu s, as explained in the ﬁrst parag r aph of the pro of, it is s uﬃcien t to pro ve that if a set Z intersects every Γ( y ) then dim H Z > d − 1. This is easily seen to hold if we can construct an o n to Lipschitz map ϕ : [ G → C \ Q that is co nstant o n every member of G , since Lipschitz maps do not incr e a se Haus- dorﬀ dimensio n, and dim H ( C \ Q ) > d − 1. Deﬁne ϕ ( z ) = y if z ∈ Γ( y ) , which is well-deﬁned by the disjointness of the member s of G . A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 31 Let us now prov e that this map is Lipschitz. Let y , y ′ ∈ C \ Q , z ∈ Γ( y ), and z ′ ∈ Γ( y ′ ). Cho ose l ∈ N + such that 1 N l < | y − y ′ | ≤ 1 N l − 1 . Then us ing N ≥ 6 we obtain | y − y ′ | > 6 N l +1 , thus, as ab ov e, there is a walk of level l + 1 s e pa rating z and z ′ . Therefor e | z − z ′ | ≥ 1 N l +1 , a nd hence | ϕ ( z ) − ϕ ( z ′ ) | = | y − y ′ | ≤ 1 N l − 1 = N 2 1 N l +1 ≤ N 2 | z − z ′ | , therefore ϕ is Lipschitz with Lipschitz constant at mos t N 2 . T o ﬁnish the pr o of, let n b e giv en a s in T he o rem 6 .2 a nd pick k ∈ N so lar ge that N = n 2 k satisﬁes (6.6). If p > p ( N ) then using Lemma 6.1 3 we deduce that P  dim tH M ( p,N ) > d  ≥ P  M ( p,N ) is full  > 0 , which implies p ( d,N ) c < 1. Iterating k times L emma 6 .6 we infer p ( d,n ) c < 1. Now, if p > p ( d,n ) c then P  dim tH M ( p,n ) > d    M ( p,n ) 6 = ∅  ≥ P (dim tH M ( p,n ) > d ) > 0 . Combining this with Lemma 6.5 we deduce that P  dim tH M ( p,n ) > d    M ( p,n ) 6 = ∅  = 1 , which co mpletes the pro of o f the theore m.  Remark 6.16 . It is well-known a nd no t diﬃcult to s ee that lim p → 1 P ( M ( p,n ) = ∅ ) = 0. Using this it is an easy consequence o f the previous theo rem that for every integer n > 1, d < 2 and ε > 0 ther e ex ists a δ = δ ( n,d,ε ) > 0 such that for all p > 1 − δ P  dim tH M ( p,n ) > d  > 1 − ε. 6.3. The upp e r esti mate of dim tH M . The argument of this s ubs ection will rely on so me ideas from [5 ]. Theorem 6.1 7. If p > 1 √ n then almost sur ely dim tH M ≤ 2 + 2 log p log n . Pr o of. A segment is called a b asic se gment if it is of the form  i − 1 n k , i n k  × { j n k } or { j n k } ×  i − 1 n k , i n k  , whe r e k ∈ N + , i ∈ { 1 , ..., n k } and j ∈ { 1 , ..., n k − 1 } . It suﬃces to show that for every basic segment S and for every ε > 0 there exists (almost sur ely , a random) a rc γ ⊆ [0 , 1] 2 connecting the endp oints of S in the ε -neighbo rho o d of S such that dim H ( M ∩ γ ) ≤ 1 + 2 log p log n . Indeed, we can almost sur ely construct the analo gous a r cs for all basic s egments, and hence obtain a basis of M cons is ting o f ‘approximate squares ’ whose b oundaries ar e of Hausdo rﬀ dimension at mos t 1 + 2 log p log n , therefor e dim tH M ≤ 2 + 2 log p log n almost s urely . Let us now c onstruct such an a rc γ for S and ε > 0. W e may assume that S is horizontal, hence it is of the form S =  i − 1 n k , i n k  × { j n k } fo r some k ∈ N + , i ∈ { 1 , ..., n k } and j ∈ { 1 , ..., n k − 1 } . W e divide S into n subseg men ts of leng th 1 n k +1 , a nd we ca ll a subseg men t  m − 1 n k +1 , m n k +1  × { j n k } b ad if bo th the adjacent squares  m − 1 n k +1 , m n k +1  × [ j n k − 1 n k +1 , j n k ] and  m − 1 n k +1 , m n k +1  × [ j n k , j n k + 1 n k +1 ] ar e in M k +1 . Otherwise we say tha t the subs e g- men t is go o d . Let B 1 denote the unio n o f the bad segments. Then inside e v ery ba d 32 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S ( i n k , j n k ) ( i − 1 n k , j n k ) Figure 5. Construc tio n of the arc γ connecting the endp o in ts of S segment w e rep eat the same pr o cedure, and o bta in B 2 and so o n. It is ea s y to see that this pro cess is (a scaled copy of ) the 1-dimensiona l fractal p erco lation with p replaced by p 2 . Let B = T l B l be its limit se t. Then by Rema rk 6 .4 (note that p 2 > 1 n ) we obtain dim H B = 1 + log p 2 log n = 1 + 2 log p log n or B = ∅ almo st sur ely . So it suﬃces to co nstruct a γ connecting the endp oints of S in the ε -neighbo r ho o d o f S such that γ ∩ M = B (except p erhaps some endp oints, but all the endp oints form a co untable set, hence a s et of Hausdorﬀ dimension 0). (See Figure 5.) But this is easily do ne. Indeed, for ev ery g o od subsegmen t I let γ I be an a rc connecting the endp oint s of I in a s mall neighbor hoo d o f I such that γ is disjoint from M apar t from the endp o in ts (this is p ossible, since either the top or the b ottom square was erase d from M ). Then γ =  S I is goo d γ I  ∪ B works.  Using Remarks 6 .3 and 6.4 this easily implies Corollary 6.18 . Almo st sur ely dim tH M < dim H M or M = ∅ . Remark 6.19. Calculating the ex act v alue of dim tH M seems to b e diﬃcult, since it w ould provide the v alue o f the critica l pr o bability p c of Chay es, Chay es and Durrett (where the phase tra nsition o ccurs, see ab ov e), a nd this is a long-sta nding op en problem. 7. Applica tion I I: The Hausdorff dimension o f the level sets of the generic co n tinuous function Now w e re tur n to Pro blem 1.3. The main goal is to ﬁnd analog ues to K irchheim’s theorem, that is, to determine the Hausdorﬀ dimension o f the le vel sets of the generic contin uous function deﬁned on a compact metr ic spa ce K . Let us ﬁrst no te that the ca se dim t K = 0, that is, when ther e is a bas is consisting of clo pen sets is trivial b ecause o f the follo wing well-known and easy fact. F or a short pro of see [1, Lemma 2.6 ]. F act 7.1. If K is a c omp act metric sp ac e with dim t K = 0 then t he generic c on- tinuous fun ction is one-t o-one on K . Corollary 7.2. If K is a c omp act metric sp ac e with dim t K = 0 then every non- empty level set of the generic c ont inuous function is of Hausdorﬀ dimension 0 . A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 33 Hence from now on we can restrict our atten tion to the case of p ositive top olog ical dimension. In the ﬁrst part of this section we prove Theore m 7.1 2 and Coro llary 7.14, our main theo r ems concer ning level sets of the ge neric function deﬁned on an arbitrary compact metric spa ce, then we us e this to der ive conclusions ab out ho mogeneous and self-s imila r spac e s in Theorem 7.15 and Co rollary 7.17. 7.1. Arbitrary compact metric spaces. The go al o f this subsection is to prov e Theorem 7.12. In or der to do this we will need tw o equiv a len t deﬁnitions of the top ological Haus do rﬀ dimension. Let us ﬁx a compa c t metric space K with dim t K > 0, and let C ( K ) denote the space of contin uous rea l-v alued functions eq uipp ed with the supremum norm. Since this is a complete metr ic spa ce, we can us e B aire catego ry arguments. Deﬁnition 7.3. Deﬁne P l = { d ≥ 1 : ∃ G ⊆ K such that dim H G ≤ d − 1 and the g eneric f ∈ C ( K ) is one-to -one on K \ G } . Deﬁnition 7.4 . W e say that a contin uous function f is d -level narr ow , if there exists a dense set S f ⊆ R such tha t dim H f − 1 ( y ) ≤ d − 1 for every y ∈ S f . Let N d be the set of d -level narrow functions . Deﬁne P n = { d : N d is so mewhe r e dense in C ( K ) } . Now we r epe a t the deﬁnition of the top olog ical Hausdorﬀ dimension. Deﬁnition 7.5. Let dim tH K = inf P tH , whe r e P tH = { d : K has a basis U such tha t dim H ∂ U ≤ d − 1 for every U ∈ U } . W e ass ume tha t by deﬁnition ∞ ∈ P n , P l , P tH . Now we s how the following theorem. Theorem 7.6 . If K is a c omp act metric sp ac e with dim t K > 0 then P tH = P l = P n . Theorem 7 .6 and Corollar y 3 .8 immediately yield t wo new equiv ale nt deﬁnitions for the top olog ical Hausdorﬀ dimension. Theorem 7.7 . If K is a c omp act metric sp ac e with dim t K > 0 then dim tH K = min P l = min P n . Before proving Theor em 7 .6 we need the following well-kno wn lemma. F or the readers conv enience we give its short pr o of. Lemma 7.8. L et K 1 ⊆ K 2 b e c omp act metric s p ac es and R : C ( K 2 ) → C ( K 1 ) , R ( f ) = f | K 1 . If F ⊆ C ( K 1 ) is c o-me ager t hen so is R − 1 ( F ) ⊆ C ( K 2 ) . Pr o of. T he map R is clearly contin uous. Using the Tietze E xtension Theorem it is not diﬃcult to see that it is also op en. W e may a s sume that F is a dense G δ set in C ( K 1 ). The contin uit y of R implies that R − 1 ( F ) is a lso G δ , thus it is enough to prov e that R − 1 ( F ) is dense in C ( K 2 ). Let U ⊆ C ( K 2 ) b e non-empty op e n, then R ( U ) ⊆ C ( K 1 ) is also no n-empt y op en, hence R ( U ) ∩ F 6 = ∅ , and therefore U ∩ R − 1 ( F ) 6 = ∅ .  34 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S Next we prove Theor em 7.6. The pro o f will consist of three lemmas . Lemma 7.9. P tH ⊆ P l . Pr o of. Ass ume d ∈ P tH and d < ∞ . Let U b e a countable ba s is o f K such that dim H ∂ U ≤ d − 1 for all U ∈ U . Now the as sumption dim t K ≥ 1 and Theorem 4.4 yield d ≥ dim tH K ≥ 1. Let F = S U ∈U ∂ U . The countable s ta bilit y of Ha us dorﬀ dimension implies dim H F ≤ d − 1. Then there exists a G δ set G ⊆ K such that F ⊆ G and dim H G = dim H F ≤ d − 1. The ab ov e deﬁnitions clearly imply dim t ( K \ G ) ≤ dim t ( K \ F ) ≤ 0. As K \ G is F σ , we can choo se compact sets K n such that K \ G = S ∞ n =1 K n and K n ⊆ K n +1 for all n ∈ N + . Let F n = { f ∈ C ( K n ) : f is one- to-one } and let us deﬁne R n : C ( K ) → C ( K n ) as R n ( f ) = f | K n for all n ∈ N + . Since dim t K n ≤ dim t ( K \ G ) ≤ 0, F act 7.1 implies that the sets F n ⊆ C ( K n ) are co-meager . Lemma 7.8 yields that R − 1 n ( F n ) ⊆ C ( K ) ar e co-meager , to o. As a countable intersection of co-meager sets F = T ∞ n =1 R − 1 n ( F n ) ⊆ C ( K ) is also co- meager. C le arly , every f ∈ F is o ne-to-one o n all K n , so K n ⊆ K n +1 ( n ∈ N + ) yields that f is o ne-to-one on S ∞ n =1 K n = K \ G . Hence d ∈ P l .  Lemma 7.10 . P l ⊆ P n . Pr o of. Ass ume d ∈ P l and d < ∞ . By the deﬁnition of P l , there e x ists G ⊆ K such tha t dim H G ≤ d − 1 and for the generic f ∈ C ( K ) for a ll y ∈ R we hav e #( f − 1 ( y ) \ G ) ≤ 1. Then dim H G ≤ d − 1 and d ≥ 1 yield dim H f − 1 ( y ) ≤ d − 1, so N d is co-meag e r , thus (everywhere) dense. Hence d ∈ P n .  Lemma 7.11 . P n ⊆ P tH . Pr o of. Ass ume d ∈ P n and d < ∞ . Let us ﬁx x 0 ∈ K and r > 0 . T o verify d ∈ P tH we need to ﬁnd an o pen set U s uch tha t x 0 ∈ U ⊆ U ( x 0 , r ) and dim H ∂ U ≤ d − 1 . W e may assume ∂ U ( x 0 , r ) 6 = ∅ , otherwise we a re done. By d ∈ P n we obtain that N d is dens e in a ball B ( f 0 , 6 ε ), ε > 0. By decr e asing r if necessa r y , we may a ssume that diam f 0 ( U ( x 0 , r )) ≤ 3 ε . Then Tietze’s E xtension Theorem provides an f ∈ B ( f 0 , 6 ε ) such that f ( x 0 ) = f 0 ( x 0 ) and f | ∂ U ( x 0 ,r ) ( x ) = f 0 ( x 0 ) + 3 ε for every x ∈ ∂ U ( x 0 , r ). Since N d is dense in B ( f 0 , 6 ε ), we ca n cho ose g ∈ N d such that || f − g || ≤ ε . By the co ns truction o f g it follows that g ( x 0 ) < min { g ( ∂ U ( x 0 , r )) } . Hence in the dense set S g (see Deﬁnition 7.4) there is a n s ∈ S g such that (7.1) g ( x 0 ) < s < min { g ( ∂ U ( x 0 , r )) } . Let U = g − 1 (( −∞ , s )) ∩ U ( x 0 , r ) , then clearly x 0 ∈ U ⊆ U ( x 0 , r ). By (7.1) we hav e ∂ g − 1 (( −∞ , s )) ∩ ∂ U ( x 0 , r ) = ∅ , therefore ∂ U ⊆ ∂ g − 1 (( −∞ , s )) ⊆ g − 1 ( s ). Using s ∈ S g we infer that dim H ∂ U ≤ dim H g − 1 ( s ) ≤ d − 1.  This concludes the pr o o f of Theo rem 7.6. Now we ar e ready to describ e the Hausdorﬀ dimension of the level sets of gener ic contin uo us functions. As a lready mentioned ab ov e, if dim t K = 0 then every level set of a gener ic contin uo us function o n K consists of at most o ne p oint. A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 35 Theorem 7.12. L et K b e a c omp act metric sp ac e with dim t K > 0 . Then for the generic f ∈ C ( K ) (i) dim H f − 1 ( y ) ≤ dim tH K − 1 for every y ∈ R , (ii) for every d < dim tH K ther e exists a non-de gener ate interval I f ,d such that dim H f − 1 ( y ) ≥ d − 1 for every y ∈ I f ,d . Note that this theor em is sharp in g eneral, see the la st but o ne par agraph of the Intro ductio n. Theore m 7.12 a ctually readily follo ws from the following more precise, but slig h tly technical version. Theorem 7.13. L et K b e a c omp act metric sp ac e with dim t K > 0 . Then ther e exists a G δ set G ⊆ K with dim H G = dim tH K − 1 s u ch that for the generic f ∈ C ( K ) (i) f is one-to-one on K \ G , henc e dim H f − 1 ( y ) ≤ dim tH K − 1 for every y ∈ R , (ii) for every d < dim tH K ther e exists a non-de gener ate interval I f ,d such that dim H f − 1 ( y ) ≥ d − 1 for every y ∈ I f ,d . Pr o of. L e t us ﬁrst prove ( i ). Theorem 7.7 implies dim tH K = min P l . Thus there exists a set G ⊆ K such that dim H G = dim tH K − 1 and the gener ic f ∈ C ( K ) is one-to-one on K \ G . By ta king a G δ hu ll of the sa me Hausdorﬀ dimension we can assume that G is G δ . As dim tH K = min P l ≥ 1, w e hav e dim H G = dim tH K − 1 ≥ 0. Hence dim H f − 1 ( y ) ≤ dim H G = dim tH K − 1 for the g eneric f ∈ C ( K ) and fo r all y ∈ R . Th us ( i ) holds. Let us now pr ov e ( ii ). Let us choo se a s e quence d k ր dim tH K . Theorem 7.7 yields d k < dim tH K = min P n for every k ∈ N + . Hence N d k is nowhere dense by the deﬁnition of P n . It follows from the deﬁnition o f N d that for every f ∈ C ( K ) \ N d k there exists a non-de g enerate interv al I f ,d k such that dim H f − 1 ( y ) ≥ d k − 1 for every y ∈ I f ,d k . But then ( ii ) holds for every f ∈ C ( K ) \ ( S k ∈ N + N d k ), and this latter set is clear ly co-meag e r , which co ncludes the pro of of the theo rem.  This immediately implies Corollary 7.14. If K is a c omp act metric sp ac e with dim t K > 0 then for the generic f ∈ C ( K ) sup  dim H f − 1 ( y ) : y ∈ R  = dim tH K − 1 . 7.2. H o mogeneous and self-s imilar compact me tri c s p aces. In this subsec- tion we show that if the compact metr ic spa ce is suﬃciently homogeneous, e.g. self-similar (see [7] or [21]) then we c an say muc h more. Theorem 7.1 5. L et K b e a c omp act metric sp ac e with dim t K > 0 such that dim tH B ( x, r ) = dim tH K for every x ∈ K and r > 0 . Then for the generic f ∈ C ( K ) for the generic y ∈ f ( K ) dim H f − 1 ( y ) = dim tH K − 1 . Remark 7.16. In fact, the author s s how in [1] that the c o ndition in Theore m 7.15 is also necessary: If K is a compact metric space w ith dim t K > 0 such that dim H f − 1 ( y ) = dim tH K − 1 for the gene r ic f ∈ C ( K ) and fo r the gener ic y ∈ f ( K ) then dim tH B ( x, r ) = dim tH K for every x ∈ K and r > 0. Before turning to the pro of of this theorem we for m ulate a cor ollary . Recall that K is self-similar if there ar e injective co n tractive similitudes ϕ 1 , . . . , ϕ k : K → K 36 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S such that K = S k i =1 ϕ i ( K ). T he sets of the form ϕ i 1 ◦ ϕ i 2 ◦ · · · ◦ ϕ i m ( K ) ar e called the elementary pie c es o f K . It is easy to see that every ball in K contains an elementary piece. Mor eov er, by Cor ollary 4.8 the top ologica l Hausdor ﬀ dimension of every elementary piece is dim tH K . Hence, using monotonicity as well, we o btain that if K is self-s imilar then dim tH B ( x, r ) = dim tH K for every x ∈ K a nd r > 0. This yields the following. Corollary 7.17. L et K b e a self-similar c omp act metric sp ac e with dim t K > 0 . Then for the generic f ∈ C ( K ) for t he generic y ∈ f ( K ) dim H f − 1 ( y ) = dim tH K − 1 . Pr o of of The or em 7.15. Theorem 7 .12 implies that for the gene r ic f ∈ C ( K ) for every y ∈ R w e have dim H f − 1 ( y ) ≤ dim tH K − 1, so we o nly have to pr ov e the opp osite inequa lity . Let us consider a sequence 0 < d k ր dim tH K . F or f ∈ C ( K ) and k ∈ N + let L f ,k =  y ∈ f ( K ) : dim H f − 1 ( y ) ≥ d k − 1  . First we show that it suﬃces to co nstruct for every k ∈ N + a co-meag er set F k ⊆ C ( K ) such that fo r every f ∈ F k the set L f ,k is co-meag e r in f ( K ). Indeed, then the set F = T k ∈ N + F k ⊆ C ( K ) is co-mea ger, a nd for every f ∈ F the set L f = T k ∈ N + L f ,k ⊆ f ( K ) is also co-mea ger. Since for every y ∈ L f clearly dim H f − 1 ( y ) ≥ dim tH K − 1, this ﬁnishes the pro of. Let us now construct such an F k for a ﬁxe d k ∈ N + . Let { B n } n ∈ N be a countable basis of K consisting of closed balls, and for all n ∈ N let R n : C ( K ) → C ( B n ) b e deﬁned a s R n ( f ) = f | B n . Let us also deﬁne B n =  f ∈ C ( B n ) : ∃ I f ,n s. t. ∀ y ∈ I f ,n dim H f − 1 ( y ) ≥ d k − 1  , (where I f ,n is understo o d to b e a non-degenera te in terv al). Finally , let us deﬁne F k = \ n ∈ N R − 1 n ( B n ) . First we show that F k is co-mea ger. By o ur assumption dim tH B n = dim tH K > d k (whic h also implies dim t B n > 0 by F act 4 .1, since d k > 0), thus Theorem 7.12 yields that B n is co- mea ger in C ( B n ). Lemma 7.8 implies that R − 1 n ( B n ) is co-meag er in C ( K ) for all n ∈ N , thus F k is als o co- meager. It remains to sho w that for ev ery f ∈ F k the set L f ,k is co-meag er in f ( K ). Let us ﬁx f ∈ F k . W e will actually show that L f ,k contains an op en set in R which is a dense subse t of f ( K ). So let U ⊆ R b e an op en set in R such that f ( K ) ∩ U 6 = ∅ . It is eno ugh to prov e that L f ,k ∩ U con tains a n interv al. Since the sets B n form a basis, the contin uity of f implies that there ex is ts an n ∈ N such that f ( B n ) ⊆ U . It is easy to s ee using the deﬁnition of F k that f | B n ∈ B n , so there exis ts a non-degenera te in terv al I f | B n ,k such that for a ll y ∈ I f | B n ,k we hav e dim H f − 1 ( y ) ≥ dim H ( f | B n ) − 1 ( y ) ≥ d k − 1 . Thu s I f | B n ,k ⊆ L f ,k . Then d k − 1 > − 1 implies ( f | B n ) − 1 ( y ) 6 = ∅ for every y ∈ I f | B n ,k , thus I f | B n ,k ⊆ f ( B n ). But it fo llows from f ( B n ) ⊆ U that I f | B n ,k ⊆ U . Hence I f | B n ,k ⊆ L f ,k ∩ U and this completes the pr oo f.  A NEW FRAC T AL DIME NS ION: THE TOPOLOGICAL HAUSDORFF DIME NSION 37 8. Open Problems First let us rec all the most int eresting op en problem. Problem 5.8. Det ermine t he almost sur e top olo gic al Hausdorﬀ dimension of t he r ange of the d -dimensional Br ownian motion for d = 2 and d = 3 . Equivalently, determine the smal lest c ≥ 0 such that the r ange c an b e de c omp ose d int o a t otal ly disc onne ct e d set and a set of Hau s dorﬀ dimension at most c − 1 almost sur ely. Now we collect a few more op en problems. The ﬁrst one conce r ns a certain Darb oux pr o per t y . Problem 8 . 1. L et B ⊆ R d b e a Bor el set and 1 ≤ c < dim tH B b e arbitr ary. Do es ther e ex ist a Bor el set B ′ ⊆ B with dim tH B ′ = c ? The following pr oblem is motiv ated by the pro of of Theorem 6 .2. The idea is to lo ok for some structura l reason b ehind lar ge top ologica l Ha usdorﬀ dimension. Problem 8. 2. Is it t rue that a c omp act metric sp ac e K satisﬁes dim tH K ≥ c iﬀ it c ontains a family of dis join t non-de gener ate c ont inua su ch that e ach set me eting al l memb ers of this family is of Hausdorﬀ dimension at le ast c − 1 ? The following r emark shows that by dro pping disjointn ess the proble m b ecomes rather simple. Remark 8.3. If K is a non-empty compact metric spa ce and S is the c o llection o f subsets o f K in tersecting every non-deg enerate contin uum then dim tH K = min { dim H S + 1 : S ∈ S } . Indeed, ﬁrst let S ∈ S be arbitra ry , and w e prove that dim tH K ≤ dim H S + 1. W e may assume that S is G δ , s ince we can take G δ hu lls with the same Hausdorﬀ dimension. Then K \ S is σ -compac t. A co mpa ct subset of K \ S do es not co n tain non-degenera te contin ua b y deﬁnition, so it is totally disconnec ted, th us it has top ological dimension at most zer o. Ther efore the countable stability of to p olo gical dimension zero for closed sets [6, 1.3.1.] yields that dim t ( K \ S ) ≤ 0. Hence Theorem 3.6 implies dim tH K ≤ dim H S + 1. Now we prov e that there exis ts S ∈ S with dim tH K = dim H S + 1. Theorem 3.6 yields that there is a set S ⊆ K with dim H S = dim tH K − 1 and dim t ( K \ S ) ≤ 0 . Then K \ S c a nnot contain any no n-degenerate co n tinuu m, so S ∈ S . Notice that the a bove remark do e s not apply even to G δ subspaces of Euclidean spaces, see Example 4.19. Finally , we consider other notions of dimensio n. Problem 8. 4. What is t he right notion t o describ e the p acking, lower b ox, or upp er b ox dimension of t he level set s of the generic c ontinuous function f ∈ C ( K ) ? Remark 8.5. W e ca n analogous ly deﬁne topo logical pac king, or low er b ox, or upper b ox dimension, respec tively . How ever, one can show that these deﬁnitions and so me natural mo diﬁca tions of them do not solve the a bove problem. The reason why these co ncepts behave diﬀerently is that b ox dimensions ar e not even countable stable, and packing dimens io n do es not admit G δ hu lls: It is easy to see that every G δ hu ll of Q ha s packing dimension 1. 38 RICH ´ ARD BALKA, ZOL T ´ AN BUCZOLICH, AND M ´ AR TON ELEKE S Ac knowledgmen ts. W e are indebted to U. B. Darji and A. M´ a th ´ e for some illu- minating discussio ns, in particular to U. B. Dar ji for sug gesting ideas tha t led to the res ults of Section 3 and hence to the simpliﬁcation of cer tain pr o o fs. W e thank the a nonymous refer ee for s e veral helpful s uggestions. References [1] R. Balk a, Z. Buczolic h, M . Elekes, T op ological Hausdorﬀ dimension and level sets of generic con tin uous functions on fr actals, Chaos Solitons F r actals 45 (2012), no. 12, 1579–1589. [2] R. Balk a, ´ A. F ark as, J. M. F raser, J. T. Hyde, Dimension and measure for generic cont inuous images, Ann . A c ad. Sci. F enn. Math. 3 8 (2013), 389–404. [3] R. 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Dep ar tment of Ma thema tics, University of W ashing ton, Box 354350, Sea ttle, W A 98195-43 50, USA a nd Alfr ´ ed R ´ enyi Institute of Ma thema tics, Hunga rian Academy of Sciences, PO Box 127, 13 64 Budapest, Hungar y and Institute of M a themat ics and In- forma tics, Eszterh ´ azy K ´ arol y College, Le ´ anyka u. 4., 3300 Eg er, Hunga r y E-mail addr ess : balka @math.wash ington.edu Institute of M a thematic s, E ¨ otv ¨ os Lor ´ and Univ ersity, P ´ azm ´ any P ´ eter s. 1/c, 1117 Budapest, Hu ngar y E-mail addr ess : buczo @cs.elte.h u Alfr ´ ed R ´ enyi Institute of M a thematic s, Hungarian Academy of Sciences, PO Box 127, 13 64 Budapest, Hungar y and Institute of M a themati cs, E ¨ otv ¨ os Lor ´ and Un iversity, P ´ azm ´ any P ´ eter s. 1 /c, 1117 Budapest, Hung a r y E-mail addr ess : eleke s.marton@r enyi.mta.hu

A new fractal dimension: The topological Hausdorff dimension

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