MAPS IN DIMENSION ONE WITH INFINITE ENTROPY 1. Introduction 1.1. Background. Adler ...

18 Pages·2017·0.24 MB·English

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