MAPS IN DIMENSION ONE WITH INFINITE ENTROPY PETERHAZARD Abstract. Wegiveexamplesofendomorphismsindimensiononewithinfinite topological entropy which are H¨older and/or Sobolev of every exponent. We also give examples of endomorphisms in dimension one which are in the big andlittleZygmundclasses. 1. Introduction 1.1. Background. Adler, Konheim and McAndrew [1] defined the topological en- tropy of a continuous self-mapping of a compact metric space as an analogue of the Kolmogorov-Sinai metric entropy of a measure-preserving transformation of a measure space. Recall that the topologicalentropy of a continuous self-mapping is a non-negative real number, possibly infinite, which is invariant under topological conjugacy. In [1], an example of a map for which the topological entropy is infi- nite was given (a full shift on infinitely many symbols). (See also [13, 6].) In [14], it was shown that even on smooth manifolds there exist examples of continuous self-mappings with infinite topological entropy. In fact, a stronger statement was shown: forsmoothcompactmanifoldsofdimensiontwoorgreater,agenerichome- omorphism(with respectto the uniform topology)has infinite topologicalentropy. In contrast, self-mappings with sufficient regularity or smoothness must have finite topological entropy. More precisely [9, Theorem 3.2.9], if f is a Lipschitz self-mapping of the compact metric space (X,d) with finite Hausdorff dimension D(X), we have the following inequality h (f)≤D(X)log+Lip (f) (1.1) top d where h (f) denotes the topological entropy of f, and Lip (f) denotes the Lips- top d chitz constant of f with respect to the metric d. (See also [8, 2].) In [4], an investigation was started into what occurs between C0 and Lipschitz regularity, in the case of homeomorphisms on smooth compact manifolds. The notionof‘between’canbetakeninseveraldifferentdirections. Forcompactsubsets of the real line, for example, given 0 ≤ α < β < 1, if Cα denotes the space of α-H¨older self-maps and CZ and CLip denote the spaces of self-maps satisfying respectively the Zygmund and Lipschitz conditions then C0 ⊃Cα ⊃Cβ ⊃CZ ⊃CLip (1.2) Date:October 8,2017. 2010 Mathematics Subject Classification. Primary: 37B40;Secondary: 37E99, 46E35,26A16. Key words and phrases. Entropy,H¨olderclasses,Sobolevclasses,Zygmundclasses. Thisworkhasbeenpartiallysupportedby“ProjetoTem´aticoDinˆamicaemBaixasDimenso˜es” FAPESPGrant2011/16265-2, FAPESPGrant2015/17909-7 . 1 2 PETERHAZARD Similarly,ifDAEdenotesthespaceofcontinuousself-mapsdifferentiable(Lebesgue)- almost everywhere, AC denotes the space of continuous self-maps which are abso- lutely continuous, BV denotes the space of continuous self-maps with bounded variation, and W1,p, 1≤p≤∞, denotes the space continuous self-maps satisfying the W1,p-Sobolev condition, then C0 =UC ⊃DAE⊃BV⊃AC≃W1,1 ⊃W1,p ⊃W1,∞ ≃CLip (1.3) With some care, (most of) these regularity classes, and the associated inclusions, canbeextendedtohigherdimensions,andeventogeneralsmoothmanifolds. In[4] an investigation into which values of entropy are possible in these two families of inclusionswasinitiated. Itwasshownthatforanyα∈[0,1)andp∈[1,∞),infinite entropy was not only possible, but a generic property in certain spaces (suitably topologised) of bi-α-H¨older homeomorphisms, and of bi-(1,p)-Sobolev homeomor- phisms, on smooth manifolds of dimension two or greater. Note that the result in [4] is only for the closure of bi-Lipschitz maps in the appropriate topology. M. Benedicks asked the following question: Benedicks’ Question: Is there a mapping in the big Zygmund class with infinite topological entropy? In the little Zygmund class? Herewegiveananswertothesequestionsinthecaseofendomorphismsoncompact one-manifolds. Specifically, we will work on the closed unit interval, but the gen- eralisation to the circle case will follow immediately. (Note that homeomorphisms on compact one-manifolds must have zero entropy.) 1.2. Summary of results. First we construct examples of endomorphisms with infinite topological entropy lying in a Ho¨lder or Sobolev class. Theorem A. There exists a continuous one-parameter family of endomorphisms f ∈C([0,1],[0,1]), a∈(0,1], with the following properties a (1) for all a∈(0,1], (a) all f are topologically conjugate a (b) h (f )=+∞ top a (c) f is not expansive, h-expansive or asymptotically h-expansive a (2) for a=1, (a) f has modulus of continuity ω(t)=tlog(1/t) a (b) f is in the Sobolev class W1,p for 1≤p<∞ a (c) Lebesguemeasureisameasureofmaximalentropyforf (thoughthere a are at least countably many such measures) (3) for a∈(0,1) (a) f is Cα if and only if α≤a. a (b) f is W1,p if an only if p<(1−a)−1 a (c) Lebesgue measure is not preserved by f , but there exist measures of a maximal entropy for f which are absolutely continuous with respect a to Lebesgue. We note that, for the specific examples considered here, half of the work is already done by Morrey’s inequality: namely, if f lies W1,p then it automatically a follows that f is Cα, where α=1− 1. However, we also give an explicit proof of a p TheoremA(3)(a). Thereasonbeingthatourconstructionismadeusingpiecewise- affine horseshoes as ‘model maps’ from which the construction is made. If the 3 modelmapwhichwestartwithisHo¨lderbutnot,forinstance,differentiablealmost everywhere,then our construction and estimates still apply. Remark 1.1. Similar examples were already constructed in [5]. However, the con- struction there made determining the possible conjugacy between different f diffi- a cult. Our approach here simplifies this, while also giving the additional dynamical information in Theorem A above. Following this we also construct examples of endomorphisms with infinite topo- logical entropy satisfying the stronger Zygmund condition. Namely, the following is shown. Theorem B. There exists f ∈C([0,1],[0,1]) with the following properties (1) (a) f is not topologically conjugate to the examples in Theorem A (b) h (f)=+∞ top (c) f is not expansive, h-expansive or asymptotically h-expansive (2) f satisfies the little Zygmund condition Remark 1.2. Observe that Theorem B gives an affimative answer to Benedicks’ question stated above. 1.3. Structure of the paper. InSection1.4,wesetupnotationandterminology for the rest of the paper, and recall some basic facts. In Section 2 we give a proof of Theorem A. First we construct the one-parameter family f from which it will a be clear that properties 1(a)–1(c) of Theorem A hold for all parameters a∈(0,1]. Afterthisanelementaryproofofproperties2(a)–2(c),i.e.,fora=1,ofTheoremA is given. Following this we prove some auxiliary propositions that are then used to prove properties 3(a)–3(b). In Section 3, after recalling some basic definitions we give a proof of Theorem B. Finally, in Section 4 we end with some remarks and open problems. 1.4. Notation and terminology. Throughout this article, we use the following notation. We denote the Euclideannorm in R by |·|R or |·| when there is not risk of ambiguity. We denote the Euclidean distance by d(·,·). 1.4.1. Ho¨lder Mappings. Given a subset Ω of R, let Cα(Ω,R) denote the set of real-valued functions f on Ω satisfying the α-Ho¨lder condition d(f(x),f(y)) def [f] = sup <∞ . (1.4) α,Ω d(x,y)α x,y∈Ω;x6=y Whenthedomainoff isclearwewillwrite[f] insteadof[f] . ThesetCα(Ω,R) α α,Ω has a linear structure and [·] defines a semi-norm1, which we call the Cα-semi- α,Ω norm. Consequently def kfkCα(Ω,R) = kfkC0(Ω,R)+[f]α,Ω (1.5) defines a complete norm on Cα(Ω,R). 1Thisalsoinducesapseudo-distance whichwewillcalltheCα-pseudo-distance. 4 PETERHAZARD 1.4.2. Sobolev Mappings. Given an open subset Ω of R, the Sobolev class W1,p(Ω) consistsofmeasurablefunctions f: Ω→R for whichthe firstdistributionalpartial derivative is defined and belongs to Lp(Ω). Then W1,p(Ω) is a Banach space with respect to the norm kuk1,p =kukLp+kDukLp (1.6) Define the space W1,p(Ω,R)=W1,p(Ω,R)∩C0 Ω,R (1.7) For f ∈W1,p(Ω,R) define (cid:0) (cid:1) 1 p [f] = |Df(x)|pdx (1.8) W1,p,Ω (cid:18)Z (cid:19) Ω Observe that W1,p(Ω,R) is a linear space and that [·] defines a semi-norm W1,p,Ω which we call the W1,p-semi-norm. Setting kfkW1,p(Ω,R) =kfkC0(Ω,R)+[f]W1,p,Ω (1.9) this defines a norm on W1,p(Ω,R) which is complete, and thus W1,p(Ω,R) is en- dowed with the structure of a Banach space. 1.4.3. Topological Entropy and Expansivity. Let (X,d) be a compact metric space. Let f be a continuous self-map of (X,d). For each n ∈ N define the distance function df(x,y)= max d(fk(x),fk(y)) (1.10) n 0≤k<n Given sets E,F ⊂X, we say that the set E (n,δ)-spans the set F with respect to f if for any x∈F, there exists y ∈E such that df(x,y)<δ. Let n r (n,δ;F)=min #E | E (n,δ)-spans F with respect to f (1.11) f (cid:8) (cid:9) (Note that: (i) if F is compact then r (n,δ;F)<∞; (ii) r (n,δ;F) increases as δ f f decreases.) For each compact set K ⊂X, define2 1 r (δ;K)=limsup logr (n,δ;K) (1.12) f f n n→∞ and h(f;K)= limr (δ;K) (1.13) f δ→0 Since X is compact we can define h(f,δ)=h(f,δ;X) (1.14) The topological entropy of f is defined by h (f)= limh(f,δ)=suph(f;K) (1.15) top δ→0 K For each ǫ>0 and x∈X define Φ (x)= f−nB (fn(x))={y :d(fn(x),fn(y))≤ǫ, ∀n≥0} (1.16) ǫ ǫ n\≥0 Recallthatf isexpansiveifthereexistsǫ>0withthefollowingproperty: given any x,y ∈X, if d(fk(x),fk(y))<ǫ, for all k ∈N, then x=y. Define h∗(ǫ)= suph(f;Φ (x)) (1.17) f ǫ x∈X 2HerewedepartfromthenotationoriginallyduetoBowen[3]. 5 Thenf ish-expansiveifh∗(ǫ)=0forsomeǫ>0;andisasymptotically h-expansive f if lim h∗(ǫ)=0. ǫ→0 f 2. Examples in Ho¨lder and Sobolev classes. Weconstructafamilyofendomorphismsf oftheunitinterval,dependingupon a the parameter a ∈(0,1], such that each f satisfies h (f )=∞, it is not expan- a top a sive, h-expansive or even asymptotically h-expansive, and such that all the f are a topologically conjugate. The main part of the work will then be in showing each f has some intermediate regularity between C0 and Lipschitz. a Remark2.1. Oncompact one-manifolds, atheorem of Misiurewicz [11] statesthat positive topological entropy, and thus infinite topological entropy, must come from some iterate possessing a horseshoe. More precisely, if h (f)>0, then there exist top sequences of positive integers kn and sn such that, for each n, fkn possesses an s -branched horseshoe3 and n 1 lim logs =h (f) (2.1) n top n→∞kn Thus examples given below, which are constructed so that certain iterates possess horseshoes, are somehow indicative of the general case. Itwillbe usefultofirstconsideranauxiliaryfamilyg ofintervalmapsdefined a,b as follows. First fix a positive integer b. Given an arbitrary interval J, let A J denotetheuniqueorientation-preservingaffinebijectionfromJ to[0,1]. Subdivide theinterval[0,1]intobclosedintervalsJ ,J ,...,J ofequallength,ordered b,0 b,1 b,b−1 from left to right. Let A = A for each k = 0,1,...,b−1. Let ν denote b,k Jb,k the unique orientation-reversing affine bijection of [0,1] to itself, For each k = 0,1,...,b−1, define g (x)=νk◦A (x), ∀x∈J . (2.2) 1,b b,k b,k More explicitly bx−k x∈J , k even g (x)= b,k (2.3) 1,b (cid:26) (k+1)−bx x∈Jb,k, k odd Observethatg iscontinuouson[0,1]. Also,[0,1]possessesag -invariantsubset 1,b 1,b on which g it acts as the unilateral shift on b symbols. In fact, h (g )=logb 1,b top 1,b (see e.g. [9, Section 3.2.c]). Next, take a continuous one-parameter family ϕ , a ∈ (0,1], of orientation- a preserving homeomorphisms of [0,1], with ϕ =id, and define 1 g =ϕ ◦g ◦ϕ−1 (2.4) a,b a 1,b a Forexample,wecouldtakeϕ equaltoq (x)=xa,thepowerfunctionofexponent a a a. (Observe that in this case g is Ca but not Cα for any α > a, provided that a,b b≥2.) Then g is continuous on [0,1]. As topological entropy is invariant under a,b topological conjugacy, we also have h (g ) = logb, for each a ∈ (0,1] and each top a,b positive integer b. We call b the number of branches of g and a the order of a,b singularity. 3Amapg possessesans-branched horseshoe ifthereisanintervalJ withspairwisedisjoint subintervalsJ1,J2,...,Js,suchthatg(Jj)⊆J forj=1,2,...,s. 6 PETERHAZARD We now define the family f as follows. For each positive integer n define the a interval I =(2−n,2−n+1] and let f be given by n a A−1◦g ◦A (x) x∈I , n=1,2... fa(x)=(cid:26) 0In a,2n+1 In x=0n (2.5) Observe that, since g fixes the endpoints of [0,1] and is continuous, the map a,2n+1 f is also continuous. Also, since, for each fixed b, all the functions g , a ∈ (0,1] a a,b are topologically conjugate, it follows that all the functions f , a ∈ (0,1], are also a all topologically conjugate. Namely, f =ψ−1◦f ◦ψ where a a 1 a ψ (x)=A−1◦ϕ ◦A (x) ∀x∈I , ∀n∈N (2.6) a In a In n NoticethattheclosureofeachintervalI istotallyinvariant. Sincethetopological n entropy of a map is the supremum of the topological entropy of its restriction to allclosedinvariantsubsets, since topologicalentropy is invariantunder topological conjugacy (see e.g. [9, Section 3.1.b]) and, as was stated above, h (g ) = logb top a,b for all b, it follows that h (f ) ≥ suph (f | ) = suph (g )=+∞ . (2.7) top a top a In top a,2n+1 n n Next, observe that, as f has arbitrarily small invariant subsets (namely the inter- a vals I ) the function f cannot be expansive. In fact, since h(f ,I )=log(2n+1) n a a n for each n, it follows that limh∗ (ǫ) ≥ lim h(f ;I ) = +∞ (2.8) ǫ→0 fa n→∞ a n Thusf isneitherh-expansivenorasymptoticallyh-expansive. Thereforeproperties a 1(a)–1(c) of Theorem A hold. Remark 2.2. That f is not asymptotically h-expansive could also be shown using a topologicalconditionalentropyinthefollowingway. By[10,Proposition3.3]infinite topological entropy h (f ) implies infinite topological conditional entropy h∗(f ). top a a However, by [10, Corollary 2.1(b)] f is asymptotically h-expansive if and only if a h∗(f )=0. a Proof of Theorem A 2(a)–2(c). For each positive integer n, define the subintervals I = A−1(J ) of I for k = 0,1,...,2n. These denote the maximal closed n,k In 2n+1,k n subintervals of I on which f is affine. n (a) Take distinct points x,y ∈[0,1]. There are three cases to consider. (x∈I ,y ∈I , n>m): SinceI andI arebothf-invariant,f(x)∈I and n m m n n f(y)∈I . Moreover,observe that m |f(x)−f(y)|≤|2−n−2−m+1|<2−m+1 (2.9) together with |x−y|≥|2−n+1−2−m|≥2−m−1 (2.10) implies that |f(x)−f(y)| 2−m+1 4 ≤ = . (2.11) ω(|x−y|) 2−m−1log2m+1 (m+1)log2 7 f(x) 0 x Figure 1. The graph of a Ho¨lder interval endomorphism with infinite topological entropy. (x=0,y ∈I ): Applying the same argument as in the previous case and m observing that f(x)=x=0 we find that |f(x)−f(y)| 2−m+1 4 2 ≤ = ≤ . (2.12) ω(|x−y|) 2−m−1log2m+1 (m+1)log2 log2 (x∈I ,y ∈I , n=m): If x and y do not lie in the same branch of f| , n m Ik then there exists y′, in the same branch as x, satisfying f(y) = f(y′) and |x−y|>|x−y′|. Moreover, |I | 1 |x−y′|≤ m = . (2.13) 2m+1 2m(2m+1) Thus |f(x)−f(y)| |f(x)−f(y′)| (2m+1)|x−y′| ≤ = (2.14) ω(|x−y|) ω(|x−y′|) |x−y′|log(|x−y′|−1) 2m+1 ≤ (2.15) log2m(2m+1) 2 ≤ +1 . (2.16) log2 In each of these cases, for x,y ∈[0,1],x6=y, |f(x)−f(y)| 2 ≤ +1 (2.17) ω(|x−y|) log2 and hence f has modulus of continuity ω, which completes the proof of part (i). 8 PETERHAZARD (b) Observe that f is differentiable except at the endpoints of the intervals I . k,l Hence |I | |f′| |= n =2n+1 . (2.18) In |I | n,k Therefore, as the I form a measurable partition of [0,1], n ∞ ∞ |f′|pdx= |f′|pdx= (2n+1)p dx (2.19) Z Z Z [0,1] nX=1 In nX=1 In ∞ = (2n+1)p2−k (2.20) nX=1 ∞ ≤ np2−(n−1)/2 (2.21) nX=1 ∞ =21/2 np2−n/2 . (2.22) nX=1 However, this last series is finite. This follows, for example, since np2−n/2 < n−2 for all n sufficiently large and by recalling that ∞ n−2 <∞. Consequently the n=1 Sobolev norm of f is finite and hence f ∈W1,p(P[0,1]). (c)Firstnotethatasg preservesLebesguemeasureµforeachb,itfollowstrivially 1,b that Lebesgue measure is invariant under f . Since h (g )=logb, it also follows 1 µ 1,b that Lebesgue measure is a measure of maximal entropy. Hence the theorem is shown. (cid:3) For each positive integer n, by performing the same construction as above but just on the union of the intervals I ,I ,... we also get the following corollary. n n+1 Corollary 2.1. There exists a sequence f ∈ C0([0,1],[0,1]) satisfying properties n 2(a)-2(c)inTheoremAaboveandwiththeadditional propertythatlim f =id n→∞ n where convergence is taken • in the Cα-topology for any α∈(0,1), • in the W1,p-topology for any p∈[1,∞). Werecallthatmapswithmodulusofcontinuitytlog(1/t)areintheHo¨lderclass Cα for every α ∈ [0,1), but they are not necessarily Lipschitz. Moreover, the map f is a Cα-limit of piecewise-affine maps. Hence it lies in the Cα-boundary 1 of the space of Lipschitz maps. When a 6= 1, the map f does not satisfy this a property. The proof of Theorem A 3(a)–3(b) could be made using the argument presented above in the proof of properties 2(a)-2(b) of Theorem A. However, we give a different proof below. For that we need the following. Proposition 2.1 (Gluing Principle). Let ω be a continuous, monotone-increasing function, locally concave at ω(0) = 0. Let f be a continuous self-mapping of the compact interval I. Let I ,I ,... denote a collection of closed intervals with pair- 1 2 wise disjoint interiors, covering I, and with the property that f| has modulus of Ik continuity ω, for all k. Let C denote the ω-semi-norm of f| . If k Ik (i) ∞ C < ∞ then f has modulus of continuity ω with ω-semi-norm k=1 k Pbounded by C = ∞k=1Ck. P 9 (ii) sup C <∞ and f| =id for all k, then f has modulus of continuity ω k k ∂Ik with ω-semi-norm bounded by C = diam(I) +2sup C . ω(diam(I)) k k Proof. For notationalsimplicity, assume that the intervals I are orderedfrom left k to right. This does not affect the proof, but simplifies indexing. Case (i). Take x,y ∈I. Assume that x<y. Then there exist integers m<n such that x∈I ,y ∈I . Consequently m n x =x<x <...<x <y =x , (2.23) m m+1 n n+1 where the points x ,x ...,x denote the left endpoints of the respective m+1 m+2 n intervals I ,I ,...,I . Let C denote the ω-semi-norm of f| , that is m+1 m+2 n k Ik d(f(z),f(w)) C = sup . (2.24) k ω(d(z,w)) z6=w∈Ik It follows that n n d(f(x),f(y))≤ d(f(x ),f(x ))≤ C ω(d(x ,x )) . (2.25) k k+1 k k k+1 kX=m kX=m However,since Λ is concave,Jensen’s inequality implies that n C ω(d(x ,x )) n C d(x ,x ) k=m k k k+1 ≤ω k=m k k k+1 (2.26) P n C (cid:18)P n C (cid:19) k=m k k=m k P P n d(x ,x ) ≤ω max C · k=m k k+1 (2.27) (cid:18)m≤k≤n k Pmaxm≤k≤nCk (cid:19) =ω(d(x,y)) . (2.28) where, for the last equality we have used that the points x are in the real line, k placedinincreasingorder. Combining inequalities (2.25) with (2.28)together with the hypothesis that ∞ C < ∞ gives the result by taking the supremum over k=1 k all possible x and y.P Case (ii). Take x, y and x ,...,x as before. Then m+1 n d(f(x),f(y))≤d(f(x),f(x ))+d(f(x ),f(x ))+d(f(x ),f(y)) (2.29) m+1 m+1 n n ≤C ω(d(x,x ))+d(x ,x )+C ω(d(x ,y)) (2.30) m m+1 m+1 n n n diam(I) ≤ 2supC + ω(d(x,y)) . (2.31) k (cid:18) ω(diam(I))(cid:19) k As this holds for all x and y, it follows that f has modulus of continuity ω, with ω-semi-norm bounded by 2sup C +diam(I)/ω(diam(I)), as required. (cid:3) k k Lemma 2.1 (Auxiliary Lemma). Let g be defined as above, where ϕ is an a,b a arbitrary concave, orientation-preserving homeomorphism, so that it possesses an extension to [0,1+1], which is also concave and a homeomorphism onto its image. b Then [g ] ≤[ϕ ] ·bα+1 (ϕ−1)′(t) αdt (2.32) a,b Cα,[0,1] a Cα,[0,1] Z[1b,1+1b](cid:12) a (cid:12) and (cid:12) (cid:12) p2(1−a) 1−p1 t p−1 [ga,b]pW1,p,[0,1] ≤[ϕa]W1,p,[0,1]·bpZ (cid:12)g (t)(cid:12) dt . (2.33) [0,1](cid:12) 1,b (cid:12)  (cid:12) (cid:12)  (cid:12) (cid:12) 10 PETERHAZARD Remark 2.3. As ϕ is monotone increasing it follows by Lebesgue’s Last Theo- a rem that is is differentiable Lebesgue-almost everywhere (see, e.g. [12]). Since it is concave it follows from Alexandrov’s theorem that is it also twice-differentiable Lebesgue-almost everywhere [7, Section 6.4]. Proof. Beforestartingthe proof,weintroducethe followingnotationandmakethe following comments. For any t∈[0,1] we use the notation t′ =ϕ−1(t) , t′′ =g (ϕ−1(t)) . (2.34) a 1,b a First consider the Ho¨lder estimate. Take k ∈ {0,1,...,b−1}. Let x,y ∈ J be b,k arbitrary distinct points. Then, by telescoping the a-H¨older difference quotient, and observing that g is affine, we find that 1,b |g (x)−g (y)| |ϕ (x′′)−ϕ (y′′)| |ϕ−1(x)−ϕ−1(y)| α a,b a,b =bα a a a a . (2.35) |x−y|α |x′′−y′′|a (cid:18) |x−y| (cid:19) Observe that x′′ and y′′ take values throughout [0,1]. Therefore |ϕ (x′′)−ϕ (y′′)| a a ≤[ϕ ] . (2.36) |x′′−y′′|α a Cα,[0,1] Next, trivially x and y take values throughout J = [k,k+1]. Therefore, since b,k b b the function ϕ−1(t) is convex and increasing on the positive real line (and thus a difference quotients on J are maximised by the derivative at the right endpoint b,k ∂+J ), b,k |ϕ−1(x)−ϕ−1(y)| (ϕ−1)′(∂−J )≤ a a ≤(ϕ−1)′(∂+J ) . (2.37) a b,k |x−y| a b,k Consequently,byProposition2.1(i),togetherwiththefactthat(ϕ−1)′ isincreasing a on the positive real line (so (ϕ−1)′ is minimised on J by its value at the left a b,k endpoint ∂−J = k) we have b,k b b−1 [ga,b]Cα,[0,1] ≤ [ga,b]Cα,Jb,k (2.38) kX=0 b−1 ≤[ϕ ] ·bα+1 (ϕ−1)′ k+1 α· 1 (2.39) a Cα,[0,1] a b b Xk=0(cid:12) (cid:0) (cid:1)(cid:12) (cid:12) (cid:12) b−1 =[ϕ ] ·bα+1 (ϕ−1)′ ∂−J α·|J | (2.40) a Cα,[0,1] a b,k+1 b,k+1 Xk=0(cid:12) (cid:0) (cid:1)(cid:12) (cid:12) (cid:12) ≤[ϕ ] ·bα+1 (ϕ−1)′(t) α dt . (2.41) a Cα,[0,1] Z[1b,1+1b](cid:12) a (cid:12) (cid:12) (cid:12) Next, consider the Sobolev case. Observe that g is differentiable everywhere a,b exceptafinitesetofpoints. Moreprecisely,g hasbreaksatexactlytheendpoints a,b ofϕ (J )fork =0,1,...,b−1. Bythechainrule,atLebesguealmosteverypoint a b,k x we have |g′ (x)|=|ϕ′(g (ϕ−1(x))||g′ (ϕ−1(x))||(ϕ−1)′(x)| (2.42) a,b a 1,b a 1,b a a |g (ϕ−1(x))| a−1 =b 1,b a . (2.43) (cid:18) |ϕ−1(x)| (cid:19) a