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Abstract: Consider d disjoint closed subintervals of the unit interval and consider an

arXiv:math/9204241v1 [math.DS] 20 Apr 1992Cantor sets in the line:scaling function and the smoothness of the shiftmapby F.Przytycki and F. TangermanAbstract: Consider d disjoint closed subintervals of the unit interval and consider anorientation preserving expanding map which maps each of these subintervals to the wholeunit interval. The set of points where all iterates of this expanding map are defined is aCantor set.

Associated to the construction of this Cantor set is the scaling function whichrecords the infinitely deep geometry of this Cantor set. This scaling function is an invariantof C1 conjugation.

We solve the inverse problem posed by Dennis Sullivan: given a scalingfunction, determine the maximal possible smoothness of any expanding map which producesit.Consider the space Σd = { 1, .., d }IN, with its standard shiftmap σσ(α1α2..) = (α2...)Denote by σ−1ithe d right-inverse of σ:σ−1i(α1α2..) = (iα1α2..)Our convention will be not to use separating comma’s in strings of symbols.Σd with the product topology is a Cantor set. Consider an embedding h of the spaceΣd = { 1, .., d }IN into IR with the standard order:h(α) > h(β) iffαm > βmwhere m is the first integer for which αm ̸= βm.

The image of h is also a Cantor set. Denoteby f the induced shiftmap on the image of h and by f−1ithe d right-inverses of f. Letr > 1.

We say that h is Cr if each of the right-inverses f−1ihave Cr extensions to IR whichare contractions. We say then that the Cantor set is Cr.Every C1+ǫ Cantor set has a scaling function, defined below and there is a simple char-acterization of those functions which are scaling functions for some C1+ǫ Cantor set.

In thispaper we describe those scaling functions which actually have to Ck+ǫ realizations. Here kis any integer greater or equal to 1 and 0 < ǫ ≤1.

We follow the convention that ǫ = 1means a Lipschitz condition.The theory for r = 1 + ǫ is essentially due to Feigenbaum and Sullivan who introducedthe scaling function. It is defined in the following manner.

Given an embedding h, thenthe shiftmap allows a canonical definition of the image of h as an intersection of nestedcollections of intervals. More precisely, define for any finite sequence (j1..jn) Ij1..jn as theconvex hull of h({ α : α1 = jn, .., αn = j1 }) Note the order in which the indices occur.Then for any j0, Ij0j1..jn ⊂Ij1..jn and the shiftmap maps Ij1..jn to Ij1..jn−1.

For the emptystring, I denotes the image of h. The sets thus constructed are not intervals, but actuallysmall pieces of the image of h. It is however convenient to think of them as intervals.For any subset J in the reals denote by < J > its convex hull and by |J| the length ofits convex hull. We will in the remainder always assume that < I > is the unit interval [0,1].1

Denote the set of finite strings j1, .., jn of length n by Σduald,n . The scaling function (ratiogeometry) at level n is a function Sn:S : Σduald,n→(0, 1)2d−1defined in the following manner.

For each j1..jn S(j1..jn) records the geometrical location ofthe d intervals { Ij0j1..jn }j0=1..d in Ij1..jn by the ratio’s of lengths of these d intervals (first dcoordinates) and d −1 gaps (last d −1 coordinates) to the length of Ij1,..,jn. In particularfor j0 = 1, .., d the j0 −th coordinate of S is given by the following formula:S(j1..jn)j0 = |Ij0..jn||Ij1..jn|The sum of all ratio’s of lengths equals one.

Therefore S actually takes values in the 2d −2dimensional simplex Simp2d−2 of (0, 1)2d−1 where the sum of the coordinates equals 1.Moreover lengths of intervals are determined by the scaling functions at all levels:|Ij1..jn| = Πk S(jk+1..jn)jk (1)Consider two finite sequences j = j1..jn and j′ = j′1..j′m. There is a canonical identificationbetween Ij and Ij′ defined as follows.

Let j ∩j′ be the longest string which agrees withboth the beginning of j and the beginning of j′. Then suitable iterates of the shiftmap mapIj to Ij ∩j′ respectively Ij′ to Ij ∩j′.

(see diagram)Ij′∩jրտIj′IjThe fundamental observation is that if the embedding is C1+ǫ then the identification mapis close to being linear in the following precise sense. Define the nonlinearity of a diffeomor-phism f on an interval aslogsupx,y,x ̸= yDf(x)Df(y)Then the nonlinearity of the identification map can be estimated from above in terms of thelength of the intermediary interval Ij ∩j′.

But then if j ∩j′ is long (i.e. |Ij ∩j′| small), thesubdivision of Ij is close to that of Ij′.

One concludes that there exists a uniform γ suchthat 0 < γ < 1|S(j1..jn) −S(j′1..j′m)| ≤γ♯(j ∩j′) (inequality 1) (2)Here ♯(j ∩j′) denotes the length of j ∩j′. Therefore for any infinite sequence j = (j1j2...)the scaling function S:S(j) =limn →∞S(j1..jn)is well defined and has a H¨older modulus of continuity:|S(j) −S(j′)| ≤γ♯(j∩j′)2

This scaling function is canonically defined on the dual Cantor set Σduald, whose elementsare infinite sequences (j1j2..). Each such sequences should be thought of as a prescribedsequence of inverse branches of the shiftmap.Say that a map is C1+ if it is C1 + ǫ for some ǫ.Theorem: [Sullivan] Every C1+ embedding has a H¨older continuous scaling function.The scaling function is a C1 invariant.

Every H¨older continuous function on the dual Cantorset with values in Simp2d−2 is the scaling function of a C1+ embedding.Here the H¨older continuity of the scaling function is defined with respect to a metric onΣduald:ρδ(j, j′) = exp(−δ ♯(j ∩j′)In the theorem δ (the metric on Σduald) is not specified so we cannot specify ǫ.The problem which remained was to understand which functions occur as scaling func-tions for C1+1 and higher smoothness. Here we give necessary and sufficient conditions fora function S to arise as a scaling function for a Ck + ǫ (k positive integer and 0 < ǫ ≤1)embedding.

The main observation is that given an embedding, we should be able to extendthe identification map between Ij and Ij′ to their convex hulls < Ij > and < Ij′ to be Ck+ǫclose to affine provided j ∩j′ is long. Here close to affine is measured after affinely rescaling< Ij > and < Ij′ > to the unit interval.

We refer to the process of changing the map byrescaling domain and range to the unit interval as renormalization.We will first characterize those functions which are scaling functions of C1+ǫ Cantor sets.This is a special case of the main theorem. We state it seperately because of its simpler form.Given a function S : Σduald→Simp2d−2.

We replace an arbitrary metric ρδ on Σdualdwitha metric ρS so that for an embedding with S as scaling function there exits K so that forevery j, j′:1K ≤|Ij ∩j′|ρS(j, j′) ≤K (3)This metric is defined as:ρS(j, j′) = supw Πn=♯(j∩j′)t=1S(jt+1jt+2..jnw)jt(3) holds by (1) because any infinite tail w changes the product by a uniformly boundedfactor (by (2)).Theorem 1: Fix 0 < ǫ ≤1. The following are equivalent:1.

There exists a C1+ǫ embedding with scaling function S.2. S is Cǫ on (Σduald, ρS).

(Here C1 means Lipschitz).Proof: That 1. ⇒2.

follows when one observes that a stronger form of (2) holds:|S(j1..jn) −S(j′1..j′n)| ≤K |Ij ∩j′|ǫ (4)3

This inequality carries over to the scaling function. Next apply (3).That 2.

⇒1., i.e. the construction of a C1+ǫ Cantor set will be done in the proof ofthe Main Theorem.Example 1: For every 0 < ǫ1 < ǫ2 ≤1 there exists S admitting a C1+ǫ1 embedding butnot C1+ǫ2.

We find it as follows: Fon an arbitrary 0 < ν < ǫ2−ǫ12we can easily find a functionS to Simp2d−2 which is C1+ǫ1+ν but not C1+ǫ2−ν on Σduald,nwith a standard metric ρδ, δ >log d. We can find in fact S so that for every j ∈Σduald,n , i = 1, .., d |−log S(j)i/δ −1| < ν/ǫ2.This is chosen so that S is Cǫ1 but not Cǫ2 with respect to the metric ρS.We now turn to the more intricate case of higher smoothness.Let A1 and A2 be two subsets of the unit interval I = [0, 1] such that both sets containthe endpoints of I and both have equal cardinality. Denote the k −th derivative operatorby Dk and denote by Dk(A1, A2) the space of Ck diffeomorphisms on I which map A1 toA2.

For every constant M > 0 consider the space of Ck-diffeomorphisms:Dkvar(M)(A1, A2) = { φ ∈Dk(A1, A2) : sup |Dkφ(x) −Dkφ(y)| < M}Lemma: Assume that A1 and A2 consist of 2d points. Assume that k < 2 d. Then foreach f, g in Dkvar(M)(A1, A2) we have for all integers t ≤k:sup |Dtf −Dtg| ≤MProof: Consider two such maps f and g. Their difference vanishes on A1.

Since 2 d > k,there exists (mean value theorem) for each t a point xt in I for which:Dtf(xt) −Dtg(xt) = 0The lemma follows by induction and integration.Given a function S as above and a point j in Σduald. Consider S(j).

It encodes a partitionof I in 2d −1 intervals. Denote by A(j) the 2 d end points of these intervals.

Consider anyj0 = 1, .., d and consider the point j0j in Σduald. Then S(j0j) specifies how the j0−th intervalin j is subdivided.

Consider two points j and j′ in ΣdualdEvery element in Dk(A(j), A(j′))maps the j0 −th interval in the domain to the j0 −th interval in the range, which we againcan renormalize. This defines a map (restrict to j0 −th interval and renormalize):Rj0 : Dk(A(j), A(j′)) →Dk({0, 1}, {0, 1})4

Main Theorem: Suppose k < 2 d. Suppose that we are given a function S as above.The following are equivalent.1. There exists a Ck+ǫ embedding with scaling function S.2.

There exists a constant C so that for all j and j′ in Σdualdand all j0 = 1, .., d:Dkvar(C(j0j, j0j′))(A(j0j), A(j0j′)) ∩Rj0(Dkvar(C(j, j′))(A(j), A(j′)) ̸= /⃝where for all j, j′ ∈Σduald,C(j, j′) = C ρS(j, j′)k+ǫ−1Discussion of statement of theorem: The statement of the theorem may appear ob-scure. We briefly discuss in an informal manner how the scaling function records smoothnessbeyond C1.1) Consider two strings j and j′ and the identification map between Ij and Ij′.

Thescalings S(j) and S(j′) record how 2 d specific points in Ij map to 2 d specific points in Ij′.Consider the renormalized identification map , and assume that we know that the variationof the kth derivative of this identification map is small.Consider any k + 1 of the 2 dspecific points. Since we know where these points map, we can compute a value of the kthderivative (just as the standard mean value theorem computes a value of the first derivativegiven 2 points and their values).

Because the variation of the k −th derivative is small , weobtain combinatorial relations between any two choices of k + 1 points. Condition 2. of thetheorem captures this idea.

It omits attempts to describe the derivative algebraically2) In fact we do not need all 2d points which appear in the definition of the ratiogeometry to be involved in the definition of Dkvar’s, k+1 would be enough (see Lemma). Inparticular for C1+ǫ the condition 2. makes impression we do not need the geometry at all.However then the condition (4)in Prof of Theorem 1 is hidden in 2. .

Without (4) a mapI →I in Dkvar(A(j0j), A(j0j′)), even linear, after renormalizing by R−1j0 may happen not tobe extendible to a map belonging to the second D in 2. .3) The condition of the main theorem seems to imply that high smoothness is notdiscussed when d is small.We can however replace d by any positive power dn in thefollowing manner.

Σd is canonically homeomorphic to Σdn, by the homeomorphism whichgroups the digits of a point in Σd in groups of n digits. This homeomorphism conjugates then −th iterate of the shiftmap on Σd to the shiftmap on Σdn.Proof of Main Theorem: We first show that 1 implies 2..

Assume that we are givena Ck +ǫ embedding h. Denote the induced shiftmap on the image by f. We may assumethat its d right-inverses extend as Ck + ǫ contractions to the unit interval, the convex hull ofthe image of h. Denote by fj′|j the identification between < Ij > and < Ij′ > and denote byFj′|j the renormalized identification defined on the unit interval J. Then fj′|j, respectivelyFj′|j, factors as a composition:fj′|j = fj′|j′∩j ◦fj′∩j|jFj′|j = Fj′|j′∩j ◦Fj′∩j|j5

Since fj′|j′∩j :< Ij′∩j > →< Ij′ > is a composition of Ck+ǫ contractions the derivatives offj′|j′∩j are controlled by the first derivative.More precisely, by a standard computation which we leave to the reader, there exists aconstant C so that for all j and j′, all 1 ≤t ≤k + ǫ|fj′|j′∩j|t ≤C |fj′|j′∩j|1Here |.|t denotes the supnorm of the t −th derivative for t integer and the α- H¨older normof the n −th derivative if t = n + α, 0 < α ≤1.But then:|Fj′|j′∩j|t = |Ij′∩j|t|Ij′||fj′|j′∩j|t≤|Ij′∩j|t−1 CThe last inequality follows because:|Ij′||Ij′∩j| = Dfj′|j′∩j(x)for some point x ∈Ij′∩j and the bounded nonlinearity of the maps.Now let j and j′ be two distinct points in Σduald. Denote by jn, respectively j′n thebeginning strings of length n. Then for n large enough j ∩j′ = jn ∩j′n and the sequenceof maps { Fj′n|j′∩j} is Ck + ǫ-equicontinuous.

Since moreover:Fj′n+m|j′∩j = Fj′n+m|j′n ◦Fj′n|j′∩jthis sequence of maps is in fact Ck+ǫ convergent. Denote by Fj′|j′∩j the limit map.

By thesame argument Fj|j′∩j is defined. Therefore: the limiting map:Fj′|j = Fj′|j′∩j ◦F −1j|j′∩jis well-defined and Ck+ǫ and therefore in Dkvar.

Since ρS(j, j′) is uniformly comparable to|Ij′∩j| we obtain that this limiting map Fj′|j in Dkvar(C′) for some uniform constant C′.Since moreover:Rj0 Fj′|j = Fj0j′|j0jwe automatically have an element in the intersection. 2. now follows.We next show that 2. implies 1..

Since S is given, we first construct an embedding ofthe Cantor set with S as scaling function. We then show that this embedding is Ck + ǫ.Fix an arbitrary infinite word w. Construct a Cantor set C in the unit interval < I > byconsecutively subdividing any interval < Ij > according to S(jw).

We obtain an embeddingwith scaling function S. Denote the induced shiftmap on the image by f0. It is defined ona Cantor set C. In order to show that this shiftmap has a Ck+ǫ extension, we verify the6

assumptions to Whitney’s extension theorem [Stein]. We will construct functions f1, ....fkon C so that for all x, y in C and l = 0, ..., k (Whitney conditions):fl(y) =t=kXt=l1(t −l)!

ft−l(x)(y −x)t−l + O(|y −x|k−l+ǫ)These functions f1, ..fk play the role of the first k derivatives of f0.The interval < I > is subdivided in d intervals < Ii >, i = 1, .., d. On each of theintervals, f0 maps Ii = C ∩< Ii > to I = C ∩J by f0. Now fix a i = 1, ..d. We willwork on each < Ii > separately.

For each t = 1, .., k define ft on Ii as:ft = limn →∞{ Dtφj1..jn,j }(j1..jn)Here φj1..jn,i : Jj1..jni →Jj1..jn is any map whose renormalization is inDvark(C(j1..jniw, j1..jnw)(A(j1..jniw), A(j1..jnw))We need to see that ft is in fact well-defined on the Cantor set. We first verify that ftis defined point wise on the Cantor set.

Consider a string j1..jn and an element j0. Forx ∈Ij0j1..jni, consider φj0j1..jni(x) and φj1..jni(x) and their t −th derivatives.Then byassumption 2. and the Lemma:|Dtφj0j1..jni(x) −Dtφj1..jni(x)| ≤|Ij0j1..jni||Ij0j1..jn|t C ρS(j1..jniw, j1..jnw)k+ǫ−1 ≤C |I(j1..jni)|k+ǫ−tTherefore we obtain the convergence on the Cantor set in fact exponentially fast.We need to check that the Whitney conditions hold on the Cantor set.

Let x and y bedistinct points in the Cantor set in < Ii >. Consider the first time that they wind up indifferent intervals in the subdivision:x ∈Jj0j1..jni, y ∈Jj′0j1....jni, j0 ̸= j′0Then again by 2.:|φj1..jni(y) −φj1..jni(x) −t=kXt=01t!

Dtφj1..jni(x)(y −x)t| ≤C |x −y|k+ǫ(and similar for the higher derivatives) where C is a uniform constant. Since |Dtφj1..jni(x) −ft(x)| ≤C |x −y|k+ǫ−t,we can take limits and obtain the Whitney conditions for the family f0, f1, ..., fk.

Conse-quently there exists a Ck+ǫ extension of f to each Ji and we have produced a Ck+ǫ embeddingof Σd with scaling function S.7

We say that two embeddings h1 and h2 are Cr equivalent if the composition h2 ◦h−11admits an extension as a C1-diffeomorphism to IR. It is well known that if h1 and h2 are Crequivalent then the composition h2 ◦h−11in fact admits an extension as a Cr-diffeomorphism.This result can also be deduced as a corollary of the method employed in the main theorem.Corollary: Assume that h1 and h2 are equivalent Ck + ǫ embeddings: h2 ◦h−11is C1.Then h2 ◦h−11is Ck+ǫ.Proof: To show that the conjugacy h2◦h−11has a Ck+ǫ extension, it suffices to constructits higher derivatives on the Cantor set and apply the Whitney extension theorem.

This canbe achieved using the same manner as that employed in the second half of the proof of themain theorem. Both embeddings have the same scaling function S so , as the embeddings areCk+1 the ratio geometries on finite levels are close to one another in the sense of condition2.

of Main Theorem.Remark: The preceding theorem is not totally satisfactory, since we do not understandhow to extract Ck-smoothness (k integer!) from the scaling function.

This is because in theprevious scheme everything which needs to be controlled is dominated by geometric series.More refined finite smoothness categories like C1+zygmund can however be treated in muchthe same way.We finally show in an example that conditions 2. of the main theorem can be explicitlychecked, by constructing for every k, d, ǫ with k < 2 d −1 and 0 < ǫ < 1 an example ofa scaling function with a Ck + ǫ realization and none of higher degree of smoothness.Example: Let Ji = [ 2 i −22 d −1, 2 i −12 d −1], i = 1, .., dDefine f : ∪i Ji →J = [0, 1] as:f(x) = A ((2 d −1) x + xk+ǫ), x ∈J1while f is affine on each Ji, i ≥2. Here the the constant A is chosen so that f(J1) = J.Of course the resulting Cantor set is Ck+ǫ.

We will show that its scalingfunction on thedual Cantor set has no Ck + ǫ1 realization for all ǫ1 > ǫ, by explicitly checking that condition2. of the main theorem does not hold for k + ǫ1.Let w be any element in Σdualdwhich does not contain the symbol 1.

Denote by 1n thestring of length n consisting of 1′s only:1n = 11..1Consider the infinite strings j = 1nw and j′ = 1n1w = 1n+1w. Consider the subdivisionA(j), respectively A(j′), of the unit interval dictated by S(j) and S(j′).

Let Φn be any map inDkvar(A(j), A(j′) for which its renormalized restriction R1Φ is in fact in Dkvar(A(1j), A(1j′).We will bound the variation of the k −th derivative of Φn from below and conclude thatcondition 2. of the main theorem is not satisfied with k + ǫ1.We denote by A(j)m the m −th point from the left in A(j). Because k < 2 d −1 thereexists x ∈[A(j)2, A(j)2d] such that:DkFj′|j(x) = Dk Φn(x)8

Recall that Fj′|j is the renormalization of fj′|j for the map f defined above. See the notationof the proof of the Main Theorem.

Similarly there exists y ∈[A(1j)1, .., A(1j)2d−1] so that:DkF1j′|1j(y) = Dk(R1Φn)(y). We have that:DkFj′|j(x) = B 2d −1−n(k−1+ǫ) xǫ(note that (2d −1)−n ∼ρS(j′, j)).The map F1j′|1j is just the renormalization of therestriction of the limit map Fj′|j to the left most interval [A(j)1, A(j)2] in the unit interval.Let y′ be the point in the interval [A(j)1, A(j)2], corresponding to y after rescaling the unitinterval back to [A(j)1, A(j)2].

Then we have that:Dk(Fj′|j)(y′) = B (2d −1)−n(k−1+ǫ) (y′)ǫwhere B is a computable constant.But |x −y′| > const (2d −1)2. Consequently:DkΦn(x) −DkΦn(y′) = const (2d −1)−n(k−1+ǫ) (xǫ −(y′)ǫand is comparable to:ρs(j′, j)k−1 + ǫi.e.

the variation of DkΦn is at least on the order of: ρs(j′, j)k−1 + ǫ.Since:limn →∞ρs(j′, j)k−1 + ǫ1ρs(j′, j)k−1 + ǫ = 0condition 2. of the theorem can not be satisfied forC(j′, j) = C ρS(j′, j)k−1 + ǫ1Reference:[Stein] Singular Integrals[Sullivan] Weyl proceedings AMS.9

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