Quasi-Minimal, Pseudo-Minimal Systems and Dense Orbits

Quasi-Minimal, Pseudo-Minimal Systems and Dense Orbits Christian Pries June 10, 2019 Abstract W e give a sho rt discuss ion ab out a weaker form of minimality (ca lle d quasi-minimality). W e call a system quasi-minimal if a ll dense orbits fo rm an open set. It is hard to find exa mples which are not already minimal. Since elliptic be haviour ma kes them minimal, these s ystems are r egarded as par ab olic s y stems. Indeed, we show that a quasi-minimal homeo mor- phism on a manifold is not expansive (hyperb olic ). 1 In tro duc tion Definition 1.1. A system ( X, T ) on a top olo gic al sp ac e X is said to b e quasi- minimal if ther e exists a non empty op en set X M ⊂ X such that for e ach x ∈ X M the orbit O ( x ) is dense in X . X M , + is the the subset of al l x ∈ X M such that O + ( x ) is dense in X . Similarly we define X M , − . We c al l ( X M ) c the exc eptiona l set. In [Gu] a sys tem is called quasi-minimal if ( X M ) c is a t most a finite set. There and in [Ka] one can fin d some examples. W e prefer the more general definition from [Ka] where these systems are regarded as parab olic systems. Indeed, we sho w that th er e is no quasi-minimal expansiv e h omeomorphism on a manifold. This theorem is w ell-kno wn for minimal homeomorphisms (see [Ma]). Our p r o of h ere is not the same, but we follo w an idea fr om [Ma]. Ma yb e Ma ˜ n ´ e knew this r esult. Before we regard expansive homeomorphisms, we need to repro of some facts kn o wn for minimal systems. Prop osition 1.2. If ( X , f ) is qu asi-minimal on a c onne cte d, p erfe ct and lo c al ly c omp act metric sp ac e X , then ( X M , f ) is total ly minimal. Fix a prime n umb er p . W e denote the elemen ts of Z p with [ q ]. F or i ∈ { 0 , 1 , . . . , p − 1 } and x ∈ X M w e set A ± [ i ] ( x ) := { y ∈ X M | ∃{ n k } k ∈ Z with pn k + i → ±∞ and f pn k + i ( x ) → y } . Lemma 1.3. 1. If x ∈ X m, ± then S i A ± [ i ] ( x ) = X M , f ( A ± [ i ] ( x )) = A ± [ i +1] ( x ) and al l A ± [ i ] ar e op en. 1 2. Given x ∈ X M and y ∈ A + [ i 0 ] ( x ) , then for al l i it holds that: A ± [ i ] ( y ) ⊂ A + [ i 0 + i ] ( x ) . Pro of: 1) The fir st t wo results are trivial. F rom Baire’s theorem we conclude that at least one A ± [ i 0 ] ( x ) h as a non empt y interior. F rom the second fact we conclude this holds for all these set, since f is a homeomorphism. 2) Cho ose, for z in A ± [ i ] ( y ), a sequence b k = pm k + i as in the definition, such that f b k ( y ) → z , and a sequence a k = pn k + i 0 as in the definition s uc h th at f a k ( x ) → y . By choosing a k gro wing fast, it follo w s that 0 < c k = a k + b k = p ( n k + m k ) + i + i 0 → ∞ and f c k ( x ) → z .  Pro of of the p rop osition: W e p r o of that ( X M , f p ) is minimal for every prime num b er, then it follo ws that ( X M , f n ) is minimal, w here n is a pro du ct of pr ime n umb ers. Fix x ∈ X M , + . Since X M is connected, we conclude f r om lemma [1.3.1] that there exists an y ∈ A + [ i 0 ] ( x ) ∩ A + [ i 1 ] ( x ) where [ i 1 ] 6 = [ i 0 ] and that for all k an d i ∈ { i 0 , i 1 } w e h a v e A ± [ k ] ( y ) ⊂ A + [ k + i ] ( x ) . F rom this and lemma [1.3.1] w e conclud e that A + [ i 0 ] ( x ) ∩ A + [ i 1 ] ( x ) m ust con tain A ± [0] ( y ). Gottsc halk sho w ed in [G] that there is n o forward or bac kw ard minimal homeomorphism on any n on compact lo cally compact space, hence w e can su pp ose th at y ∈ X M , + and tak e an k 0 suc h that x ∈ A + [ k 0 ] ( y ). Again w e conclude that for all n and i ∈ { i 0 , i 1 } we h a v e: A + [ n ] ( x ) ⊂ A + [ k 0 + n ] ( y ) ⊂ A + [ k 0 + n + i ] ( x ) . W e supp ose that [ c 0 ] := [ k 0 + i 0 ] 6 = 0 (ot h er w ise take i 1 ) and conclude, since p is a prime num b er, that for all n A + [ n ] ( x ) = A + [ n + c 0 ] ( x ) = ... = A + [0] ( x ) , whic h means the orbit of x u nder f p is dense.  Let us generalize the th eorem. Theorem 1.4. If ( X , f ) is quasi-minimal on a lo c al ly c onne cte d, p erfe ct and lo c al ly c omp act metric sp ac e X then ther e i s a finite numb er of c onne cte d c omp o- nents { C i } i ∈ I and ther e exists a k such that f k ( C i ) = C i for al l i . In p articular , k is the numb er of c onne cte d c omp onents { C i } i ∈ I and for every prime numb er p > k we have ( X M , f p ) minimal. 2 Pro of: Let { C i } i ∈ I b e the connected comp onents. Giv en x ∈ X M , + ∩ C 0 , let j ( i ) denote the uniqu e elemen t of I suc h that f i ( x ) ∈ C j ( i ) . W e conclude f i ( C 0 ) = C j ( i ) , since f is an homeomorphism. The orbit of x is dense, so there exist t wo i 0 < i 1 suc h that f i 0 ( C 0 ) = f i 1 ( C 0 ) where the distance b etw een i 0 and i 1 is minimal. S o th er e can b e only finitely many comp onents and k := i 1 − i 0 m ust b e the n umber of connected comp on ents { C i } i ∈ I . The pr o of of prop osition [1.2] works w ell in the case where p is b igger than the num b er of connected comp onents, since then there must exist an element y ∈ A + [ i 0 ] ( x ) ∩ A + [ i 1 ] ( x ) . for ev ery x ∈ X m, +  2 T op ological Dynamics Definition 2.1. A home omor phism f : X → X on a metric sp ac e ( X , d ) is c al le d exp ansive if ther e is a c onst ant e > 0 such that for x 6 = y we have sup n ∈ Z { d ( f n ( x ) , f n ( y )) } ≥ e . No w we b egin to pro of the follo wing theorem. Theorem 2.2. Su pp ose ( X , d ) is a non trivial, lo c al ly c onne cte d and c omp act metric sp ac e and ( X , f ) is exp ansive, then f is not quasi-minimal. Giv en ǫ > 0 an d x ∈ X , w e can define the lo cal stable set W s ǫ ( x ) and the lo cal uns table set W u ǫ ( x ) by W s ǫ ( x ) = { y ∈ X | d ( f i ( x ) , f i ( y )) ≤ ǫ, ∀ i ≥ 0 } , W u ǫ ( x ) = { y ∈ X | d ( f i ( x ) , f i ( y )) ≤ ǫ, ∀ i ≤ 0 } . Here are some fu ndament al theorems. Theorem 2.3 (Reddy) . If f : X → X is an exp ansive home omorphism on a c omp act metric sp ac e ( X, d ) , then ther e ar e c onstants a > 0 , b > 0 , 0 < γ < 1 and a c omp atible metric D such that for al l ǫ ≤ a we have y ∈ W s ǫ ( x ) → D ( f i ( x ) , f i ( y )) ≤ bγ i D ( x, y ) ∀ i ≥ 0 y ∈ W u ǫ ( x ) → D ( f i ( x ) , f i ( y )) ≤ bγ i D ( x, y ) ∀ i ≤ 0 Pro of: See [Re]  Let us defin e, for σ = { u, s } C σ ǫ ( x ), th e connecte d comp onent of x in W σ ǫ ( x ) and S δ ( x ) := { y | d ( x, y ) = δ } . Note that C σ ǫ ( x ) ⊂ B ( x, ǫ ). T h e follo wing holds: Theorem 2.4 (Hiriade) . If If f : X → X is an exp ans ive home omorphism on a non trivial, c onne cte d, lo c al ly c onne c te d and c omp act metric sp ac e ( X, d ) then for every ǫ > 0 ther e is an δ ( ǫ ) > 0 such tha t C σ ǫ ( x ) ∩ S δ ( ǫ ) ( x ) 6 = ∅ . 3 Pro of: See pr op osition C in [Hi].  It is easy to see that in theorem [2.4] f or all δ ≤ δ ( ǫ ) C σ ǫ ( x ) ∩ S δ ( x ) 6 = ∅ h olds . Before sho wing our theorem we need tw o usefull lemmas. Lemma 2.5. Supp ose ( X , D ) is a non trivial, c onne cte d, lo c al ly c onne cte d and c omp act metric sp ac e and ( X, f ) is an e xp ansive quasi-minimal home omor - phism, wher e D is the metric fr om the or em [2.3]. Mor e over a, b and γ ar e the c onst ants f r om the or em [ 2.3]. Th en ther e exist c onstant s ǫ > 0 , c > 0 , r > 0 and a prime inte ger p such that the fol lowing hold: 1. Y := X − B ( X c M , 2 ǫ ) c ontains an element z such that B ( z , r ) ⊂ Y 2. 0 < ǫ < a and 4 ǫ < r 3. 4 c < ǫ , 8 c < ǫ 2 b and 4 c < δ ( ǫ ) wher e δ ( ǫ ) is taken fr om the or em [2.4] 4. f p is minimal on X M 5. If Γ is a c onne cte d c omp act su b set of W s ǫ ( x ) suc h that x ∈ Γ and c ≤ D iam (Γ) ≤ 2 c , then f − p (Γ) ⊂ W s ǫ 2 ( f − p ( x )) and D iam ( f − p (Γ)) ≥ 6 c . Pro of: Ch o ose first 0 < ǫ < a so s mall that 1 and 2 is satisfied for some r > 0. Cho ose no w a large prime inte ger p suc h that 4 holds and moreo v er such that for g , h ∈ X with g ∈ W s ǫ ( h ) we hav e D ( f p ( g ) , f p ( h )) ≤ D ( g, h ) 12 (use [2.3]). Cho ose n o w a s mall c > 0 su c h that 3 holds and moreo v er, if Q is any set with D iam ( Q ) ≤ 2 c , then D ( f i ( g ) , f i ( h )) < ǫ 2 for all − p ≤ i ≤ 0 and g , h ∈ Q (use compactness). If Γ is a co nn ected compact subset of W s ǫ ( x ) s uc h that x ∈ Γ and c ≤ D iam (Γ) ≤ 2 c , then f − p (Γ) ⊂ W s ǫ 2 ( f − p ( x )). This follo ws fr om theorem [2.3], sin ce w e ha v e b y constru ction D ( f i ( g ) , f i ( h )) < ǫ 2 for all p ≥ i ≥ 0 an d g , h ∈ f − p (Γ). Moreo v er we ha ve D iam (Γ) < 2 c and therefore for all i ≥ 0 and l ∈ Γ D ( f i ( l ) , f i ( x )) ≤ bγ i D ( l , x ) < bγ i 2 c < ǫ 4 , hence D ( f i ( g ) , f i ( h )) < ǫ 2 for all i ≥ 0 and g , h ∈ Γ, th us f − p (Γ) ⊂ W s ǫ 2 ( f − p ( x )). If D iam ( f − p (Γ)) < 6 c , then D iam (Γ) < c 2 .  Lemma 2.6. Given a non trivial, c onne cte d, lo c al ly c onne cte d and c omp act metric sp ac e ( X , D ) , the nu mb ers c, ǫ as in lemma [2.5] and a c onne cte d c omp act subset Γ of W s ǫ 2 ( x ) with x ∈ Γ and Diam (Γ) ≥ 6 c , then ther e ar e two p oints α, β ∈ Γ and two c omp act c onne cte d sets α ∈ Γ α , β ∈ Γ β such that the fol lowing holds: 1. Γ α , Γ β ⊂ Γ 2. inf { d ( g , h ) | g ∈ Γ α , h ∈ Γ β } > c 2 3. Γ α ⊂ W s ǫ ( α ) and Γ β ⊂ W s ǫ ( β ) 4 4. c ≤ D iam (Γ α,β ) ≤ 2 c . Pro of: S ince Γ is compact and connected, c ho ose t w o p oint s α, β ∈ Γ suc h that d ( α, β ) = 4 c . L et Γ α b e th e connected comp onen t of α in B ( α, c ) ∩ Γ and define Γ β analogously . It is clear that 1, 2 holds and 4 follo ws f rom c ≤ δ ( ǫ ). 3 follo ws from the triangle inequalit y as in th e p r o of of [2.5].  Pro of of the th eorem: First w e kn o w that there are at most fi nitely man y connected comp onen ts, since X is locally conn ected and f is quasi-minimal. W.l.o.g f is strictly quasi-minimal, otherwise we apply [Ma]. B y applying lemma [2.4] w e c ho ose an x in B := B ( z , r ) s u c h that W := W s ǫ ( x ) lies in B and there is an op en set U with D iamU < c 2 with W ∩ U = ∅ . Moreo v er w e can assume that O + ( x ) is not d ense hence O + ( y ) is not den se for all y ∈ W and so the bac kw ard orbit of y und er f p is dense for all y ∈ W , otherwise w e c ho ose ǫ smaller. Cho ose b y applying th e previous lemmas a compact connected set Γ in W con taing x with c ≤ D iam (Γ) ≤ 2 c . Ind eed, C s ǫ in tersects S δ ( x ) for all δ ≤ δ ( ǫ ). But 4 c < δ ( ǫ ) and ther efore the connected comp onent of x (denoted b y Γ) in C s ǫ ∩ B ( x, c ) satisfies c ≤ Diam (Γ) ≤ 2 c . W e kno w from lemma [2.4 ] that f − p (Γ) ⊂ W s ǫ 2 ( f − p ( x )) and Diam ( f − p (Γ)) ≥ 6 c , so w e apply lemma [2.5] to f − p (Γ) to get a set Γ 1 with Γ 1 ⊂ f − p (Γ), x 1 ∈ Γ 1 ⊂ W s ǫ ( x 1 ) c ≤ D iam (Γ) ≤ 2 c and Γ 1 ∩ U = ∅ . Again we conclude that f − p (Γ 1 ) ⊂ W s ǫ 2 ( f − p ( x )) and D iam ( f − p (Γ 1 )) ≥ 6 c . Rep eating this construction, w e get a s equ ence of compact conn ected sets Γ i with Γ 0 = Γ, Γ i ∩ U = ∅ and f p (Γ i +1 ) ⊂ Γ i . T ak e a p oint w in T i f ip (Γ i ) ⊂ W . Th en we hav e th at O − ,f p ( w ) do es not in tersect U , hence f is n ot quasi-minimal.  One could ask why the pro of do es n ot w ork in the case where th e map is only transitiv e: W e need qu asi-minimality to ha v e W := W s ǫ ( x ) ⊂ X M . 3 Algebraic Dy namics W e mak e a sh ort excur se to algebraric dynamics. W e think this is a quite natural question ab out qu asi-minimal systems in this con text. Theorem 3.1. L et ( G, ◦ ) b e a Lie gr oup and τ an autom orphism. If ( G, τ ) is quasi-minimal, then G is trivial. Lemma 3.2. If ( G, τ ) is quasi-minimal, then G is c omp act and c onne cte d. Pro of: Aoki show ed in [A] th at an automorphism h a ving a d ense orbit on a lo cally compact m etric group X implies that X is compact. Therefore G is compact. Since e is a fi xed p oint, G is connected.  Pro of of theorem [3.1] : W.l.o.g. G is compact and connected. Let us supp ose that G is ab elian, then G is a torus. It is w ell-kno wn, th at for eve ry automorphism on G the p erio dic orbits are dense, hence G is a fi nite set, there- fore G is trivial. W e pr o of the lemma now by induction on the dimension. I t holds for dim 0 and 1. Giv en no w a non ab elian Lie group of d im G = n . τ is ergo dic by [R], since there is a dense orbit and therefore from lemma 1 in [K ] 5 w e conclude that G is nilp oten t and the center Z of G has p ositiv e dimen s ion. The induced automorphism on the lo w er dim en sional Lie group G/ Z is quasi- minimal and therefore is G = Z , so G is ab elian, h ence trivial.  References [A] N. Aoki, Dense orbits of automorphisms and compactness of group s, T op ology and its applications 20, p .1-15, 1985. [Go] W.H. Gottsc halk, Orb it-closure dec omp ositions ans almost p erio dic p oint s prop erties. Bull. Amer. Math. So c. 50, (1944), 915-919. [Gu] C. Guiterrez and B. Pires, On Pei xoto’s Conjecture F or Flo ws On Non- orien table 2-Manifolds, Pro ceeding of the American Mathematical So ci- et y , V ol.133, No.4, 1063-107 4, 2004. [Hi] K. Hiraide, Dynamical systems of expan s iv e maps on compact manifolds , Sugaku Exp ositions V ol. 5, No. 2, p. 133-154, 1992. [K] R. Kaufmann and M. Ra jogopalan, On automorph isms of a lo cally com- pact group, Mic h. Math. J . 13, 373- 374, 1966 . [Ka] A. Katok and B. Hasselblatt , Introdu ction to the Mo dern T h eory of Dynamical Systems, Cambridge Un iv ersit y P ress, Cambridge 1995 [Ma] R. Ma ˜ n ´ e , Exp ansiv e homeomorphisms and top ological d imension, T rans. Amer. Math. So c. 252, p. 313-319, 197 9. [R] M. Ra jagoplan and B.M. Sc hreib er, Ergo dic automorphisms and affine transformations, Pro c. Japan Acad., 46, p. 633-636, 197 0. [Re] W. Reddy , Expansiv e canonical co ordinates are hyperb olic, T op ology and its Appl. 15, p .205-2 10, 1983. Christian Pries F akult ¨ a t f ¨ u r Mathematik Ruhr-Universit¨ a t Bo c hum Univ ersit¨ a tsstr. 150 44780 Bo c hum German y Christian.Pries@ru b.de 6