1. Introduction
Studying the dynamical properties of discontinuous hyperbolic dynamical systems is important for understanding many relevant systems (such as billiards and optimal control theory) but it is difficult if the system is multi-dimensional. A first step towards addressing this problem can be to divide the problem into two cases: the piecewise expanding case and the piecewise contracting case. Here we address the case of piecewise contracting systems. Because the attractor can differ drastically depending on the kind of piecewise contraction, a first approach is to identify a large class of piecewise contractions that exhibit similar dynamical properties.
In recent decades, the properties of the attractor of piecewise contractions have been studied under different settings (see [Reference Bruin and Deane3, Reference Catsigeras, Guiraud and Meyroneinc7, Reference Catsigeras, Guiraud, Meyroneinc and Ugalde8, Reference Nogueira and Pires21, Reference Nogueira, Pires and Rosales22]). To begin with, Bruin and Deane (see [Reference Bruin and Deane3]) studied families of piecewise linear contractions on the complex plane
${\mathbb C}$
and proved that, for almost all parameters, each orbit is asymptotically periodic.
In the case of one-dimensional piecewise contractions, Nogueira and Pires, in [Reference Nogueira and Pires21], studied globally injective piecewise contractions on a half closed unit interval
$[0,1)$
with partition of continuity consisting of n elements and concluded that such maps have at most n periodic orbits: that is, the attractor can be a Cantor set or a collection of at most n periodic orbits or a union of a Cantor set and at most
$n-1$
periodic orbits. In particular, when the attractor consists of exactly n periodic orbits, the map is asymptotically periodic (the limit set of every element in the domain is a periodic orbit). They also proved that every such map on n intervals is topologically conjugate to a piecewise linear contraction of n intervals whose slope in absolute value equals
$1/2$
. We would like to emphasise that this result does not imply that any two piecewise contractions close enough to each other are topologically conjugate to each other. Hence, this result is not a stability result for the class of piecewise contractions. In [Reference Gaivão and Nogueira12], piecewise increasing contractions on n intervals are considered and it is proved that the maximum number of periodic orbits is n. The authors prove that the collection parameters that give a piecewise contraction with non-asymptotically periodic orbits is a Lebesgue null measure set whose Hausdorff dimension is large or equal to n. In [Reference Nogueira, Pires and Rosales22], Nogueira, Pires and Rosales proved that generically (under
${\mathcal C}^0$
topology) globally injective piecewise contractions of n intervals are asymptotically periodic and have at least one and at most n internal periodic orbits (such orbits persist under a sufficiently small
${\mathcal C}^0$
-perturbation; refer to [Reference Nogueira, Pires and Rosales22] for precise definition). In [Reference Nogueira, Pires and Rosales24], the authors prove that almost all translations within a small neighbourhood of a
$\lambda$
-affine contraction are asymptotically periodic. In [Reference Calderon, Catsigeras and Guiraud4], Calderon, Catsigeras and Guiraud proved that the attractor of a piecewise injective contraction consists of finitely many periodic orbits and minimal Cantor sets. In [Reference Janson and Oberg16, Reference Laurent and Nogueira20], the authors study symbolic coding associated to piecewise contractions on the unit interval and prove that they are related to the symbolic coding of rotations of the circle.
In higher dimensions, Catsigeras and Budelli (see [Reference Catsigeras and Budelli6]) proved that a finite dimensional piecewise contracting system with separation property (injective on the entire domain except for the discontinuity set) generically (under a topology that is finer than the topology we use for proving openness and coarser than the topology we use for proving density) exhibits at most a finite number of periodic orbits as its attractor. Here, we obtain similar, actually stronger, results without assuming the separation property.
In any (finite) dimension, in [Reference Catsigeras, Guiraud, Meyroneinc and Ugalde8], the authors show that if the set of discontinuities and the attractor of a piecewise contraction are mutually disjoint, then the attractor consists of finitely many periodic orbits. This result is a by-product of our arguments as well. In [Reference Gambaudo and Tresser14], the authors study symbolic dynamics associated to piecewise contractions (referred to as quasi-contractions in that article) and categorize its association into different kinds of circle rotations.
In the month after we submitted this article, we noticed a new preprint [Reference Gaivão and Pires13] in which the authors provide a measure-theoretical criterion for asymptotic periodicity of a parametrized family of locally bi-Lipschitz piecewise contractions on a compact metric space.
Nevertheless, the occurrence of chaotic behaviour in such systems has been addressed in the literature, and [Reference Kruglikov and Rypdal19] provides an example of a piecewise affine contracting map with positive entropy. The presence of a Cantor set in the attractor has also been studied rigorously. Examples of such maps for one dimension are given in [[Reference Catsigeras, Guiraud, Meyroneinc and Ugalde8], Example 4.3] and [Reference Coutinho9], and for three dimensions are given in [[Reference Catsigeras, Guiraud, Meyroneinc and Ugalde8], Example 5.1], where it is also proved that such a piecewise contraction turns out to have positive topological entropy. In [Reference Pires25], it is proved that, given a minimal interval exchange transformation with any number of discontinuities, there exists an injective piecewise contraction with Cantor attractors topologically semi-conjugate to it and, conversely, that piecewise contractions with Cantor attractors are topologically semi-conjugate to topologically transitive interval exchange transformations. Additionally, in [Reference Pires25] (respectively, in [Reference Catsigeras, Guiraud and Meyroneinc7]), it is proved that the complexity (the complexity function of the itineraries of orbits; refer to [Reference Catsigeras, Guiraud and Meyroneinc7, Reference Pires25]) of a globally injective piecewise contraction (respectively, piecewise contraction with separation property) on the interval is eventually affine (which is eventually constant in the case of piecewise contractions with no Cantor attractors).
The global picture presented by the above articles is that the Cantor attractors are rare, but can exist in exceptional (but constructible) cases, and many such explicit examples have been rigorously studied.
Piecewise contractions have also been used to study some models of outer billiards (see [Reference Del Magno, Gaivao and Gutkin10, Reference Gaivão11, Reference Jeong17]), where a billiard map is constructed such that it is a piecewise contraction, and so the properties of piecewise contractions are relevant in the study of such billiard maps.
Note that, in the above papers (and in this article), maps with only finitely many partition elements are considered.
The layout of this article is as follows.
Section 2 is dedicated to definitions, settings and the statement of results. In §3, we prove that the set of piecewise contractions with attractor disjoint from the set of discontinuities is open, under a rather coarse topology, and that if the maps are also piecewise injective (and hence not necessarily globally injective), then they are topologically stable. In §4, we prove that piecewise contractions with the attractor disjoint from the set of discontinuities are dense, under a rather fine topology, among the piecewise injective smooth contractions. Finally, we have three appendices collecting some needed technical facts.
2. Settings and results
Throughout this article, we work with
$(X,d_0)\subset ({\mathbb R}^d,d_0)$
, a compact subset of
${\mathbb R}^d$
, where
$d\in {\mathbb N}$
and
$d_0$
is the standard Euclidean metric on
${\mathbb R}^d$
. Under these settings, we define a piecewise contraction as follows.
Definition 2.1. (Piecewise contraction)
A map
$f:X\to X$
is called a piecewise contraction if
$\overline {f(X)}\subset \mathring {X}$
and there exist
$m\in {\mathbb N}$
and a collection
${\boldsymbol P}(f)=\{P_i: P_i=\mathring { P_i}\}_{i=1}^m$
of subsets of X such that:
-
•
$X=\bigcup _{i=1}^{m}\overline P_i$
, where
$ P_i\cap P_j=\emptyset $
whenever
$i\neq j$
; -
•
$f|_{P_i}$
is a uniform contraction, that is, there exists
$\unicode{x3bb} \in (0,1)$
such that, for all
$i\in \{1,2,\ldots ,m\}$
and for any
$x,y\in P_i$
,
$$ \begin{align*} d_0(f(x),f(y))\leq \unicode{x3bb} d_0(x,y); \end{align*} $$
-
• there exists a partition
$\{\tilde P_i\}$
,
$\tilde P_i\cap \tilde P_j=\emptyset $
for
$i\neq j$
,
$P_i\subset \tilde P_i\subset \overline {P}_i$
,
$X=\bigcup _{i=1}^{m}\tilde P_i$
such that
$f|_{\tilde P_i}$
is continuous.
Here
$\unicode{x3bb} $
is called the ‘contraction coefficient’ of f,
${\boldsymbol P}(f)$
is called the ‘partition of continuity’ and
$\overline {P_i}$
and
$\partial P_i$
represent the closure and boundary, respectively, of a partition element
$P_i$
.
Remark 2.2. The third condition pertaining to the partition
$\tilde P$
states that the values of f on the boundary of partition elements must be the limit of the values of f inside some elements, but no particular condition is imposed on which element.
For a piecewise contraction f with partition of continuity
${\boldsymbol P}(f)=\{P_1, P_2, \ldots ,P_m\}$
, we define
$\partial {\boldsymbol P}(f)$
as the boundary of the partition
${\boldsymbol P}(f)$
given by
$$ \begin{align*} \partial{\boldsymbol P}(f)=\bigcup_{i=1}^m \partial P_i. \end{align*} $$
We denote by
$\Delta (f)$
the union of the set of discontinuities of f with
$\partial X$
. To avoid confusion in the choice of partition of continuity for a given piecewise contraction, we define the following.
Definition 2.3. (Maximal partition)
A partition
${\boldsymbol P}(f)$
is called the maximal partition of a piecewise contraction f if
$\partial {\boldsymbol P}(f)=\Delta (f)$
.
Any partition of continuity of f is a refinement of the maximal partition. For a piecewise contraction with maximal partition
${\boldsymbol P}(f)=\{P_1^1,P_2^1,\ldots ,P_m^1\}$
, we define the partition
${\boldsymbol P}(f^n)= \{P_1^n,P_2^n,\ldots ,P_{m_n}^n\}$
, relative to the nth iterate
$f^n$
of f, where, for every
$k\in \{1,2,\ldots ,m_n\}$
,
$$ \begin{align}{} P_{k}^n= P_{i_0}^1\cap f^{-1}(P_{i_1}^1)\cap f^{-2}(P_{i_2}^1)\cap \cdots\cap f^{-(n-1)}(P_{i_{n-1}}^1), \end{align} $$
and where
$i_j\in \{1,2,\ldots ,m\}$
for every
$j\in \{0,1,\ldots ,n-1\}$
. Note that
${\boldsymbol P}(f)$
being the maximal partition of f does not imply that the partition
${\boldsymbol P}(f^n)$
is the maximal partition of
$f^n$
.
Remark 2.4. Throughout this article, for a given piecewise contraction f, the partition that we consider is the maximal partition
${\boldsymbol P}(f)$
, whereas for its iterates
$f^n$
we consider the partition
${\boldsymbol P}(f^n)$
, as defined in equation (2.1), which may not be the maximal partition of
$f^n$
.
One of the goals of this article is to understand the attractor of piecewise contractions. On that note, we recall the standard definition of an attractor.
Definition 2.5. (Attractor)
For a piecewise contraction f with
${\boldsymbol P}(f)=\{P_i\}_{i=1}^m$
, the attractor is defined as
$\Lambda (f)=\bigcap _{n\in {\mathbb N}}\overline {f^n(X)}$
.
When working with discontinuous maps, it is natural to talk of Markov maps (maps with Markov partitions), so we first give the following definition and then the definition of Markov maps in our settings.
Definition 2.6. (Stabilization of partition)
For a piecewise contraction f, we say that the maximal partition
${\boldsymbol P}(f)$
stabilizes if there exists
$N\in {\mathbb N}$
such that, for all
$P\in {\boldsymbol P}(f^N)$
, there exists
$Q\in {\boldsymbol P}(f^N)$
such that
$\overline {f^N(P)}\subset Q$
. We call the least such number, N, the ‘stabilization time’ of
${\boldsymbol P}(f)$
.
Definition 2.7. (Markov map)
A piecewise contraction whose maximal partition stabilizes is called a Markov map.
Now we are able to state our first result (proved in §3).
Theorem 2.8. A piecewise contraction f with
$\Lambda (f)$
as the attractor and
$\Delta (f)$
as the union of the set of discontinuities and
$\partial X$
satisfies that
$\Lambda (f)\cap \Delta (f)=\emptyset $
if and only if it is Markov. Moreover, the attractor of a Markov map consists of periodic orbits.
Remark 2.9. Note that, given a Markov map f, N its stabilization time and
${\boldsymbol P}(f^N)=\{Q_1,\ldots , Q_l\}$
the associated partition, then f induces a dynamics 
on
$\Omega (f):=\{1,\ldots , l\}$
by the rule 
. See Lemma 3.2.
To state further results, we need to add some hypotheses on our system, and thus we give the following definition.
Definition 2.10. (Piecewise injective contraction)
A piecewise contraction f with partition
${\boldsymbol P}(f)=\{P_i\}, i\in \{1,2,\ldots ,m\}$
is called a piecewise injective contraction if, for all
${i\in \{1,2,\ldots ,m\}}$
, there exists
$U_i=\mathring U_i\supset \overline {P_i}$
and an injective contraction
$\tilde f_i: U_i\to {\mathbb R}^d$
such that
$\tilde f_i|_{U_i}\supset P_i$
and
$\tilde f_i|_{P_i}=f|_{P_i}$
.
For any piecewise contractions
$f,g $
with partitions
${\boldsymbol P}(f)=\{P_1,P_2,\ldots ,P_m\}$
and
${\boldsymbol P}(g)=\{Q_1,Q_2,\ldots ,Q_m\}$
, respectively, we define
$$ \begin{align*} H({\boldsymbol P}(f),{\boldsymbol P}(g))&=\{\psi\in {\mathcal C}^0(X,X)\;:\; \psi\text{ is an homeomorphism} \\ &\qquad \text{ for all }\ P \in {{\boldsymbol P}}(f), \text{ there exists } Q \in {{\boldsymbol P}}(g): \psi(P)=Q \}. \end{align*} $$
We define the distance
$\rho $
as follows (see Lemma 3.1 for the proof that
$\rho $
is a metric).
$$ \begin{align*} \begin{aligned} &\rho(f,g):=\\ &\quad \begin{cases} \ A &\text{if } H({\boldsymbol P}(f),{\boldsymbol P}(g))=\emptyset,\\ \inf\limits_{\psi\in H({\boldsymbol P}(f),{\boldsymbol P}(g))} \{\|\psi-\mathrm{id} \|_{{\mathcal C}^0(X,X)}+\|f-g\circ\psi\|_{{\mathcal C}^0(X,X)} \} & \text{otherwise,} \end{cases} \end{aligned} \end{align*} $$
where
$A=2 \operatorname {diam}(X)$
and
$\mathrm {id}$
is the identity function, that is,
$\mathrm {id}(x)=x$
.
Evidently,
$\rho (f,g)\leq A$
for any piecewise contractions
$f,g$
. Furthermore, notice that the metric (proved in Lemma 3.1)
$\rho $
is essentially a distance between two tuples
$(f,{\boldsymbol P}(f))$
and
$(g,{\boldsymbol P}(g))$
for any two piecewise contractions
$f,g$
, where
${\boldsymbol P}(f)$
and
${\boldsymbol P}(g)$
are the maximal partitions of f and g, respectively.
For an arbitrary
$\sigma \in (0,1)$
, we define the distance
$d_1$
as
$$ \begin{align*} d_1(f,g)=\sum_{n=1}^{\infty}\sigma^{n}\rho(f^n,g^n). \end{align*} $$
Under this metric, we have the following result (proved in §3).
Theorem 2.11. The set of Markov piecewise contractions is open in the set of piecewise contractions under the metric
$d_1$
. Moreover, any two Markov piecewise contractions
$f,g$
close enough (with respect to
$d_1$
), stabilize at the same time.
To be able to state our stability result, we need to discuss the dynamics of the partition elements.
Definition 2.12. (Wandering set)
For a piecewise contraction
$f:X\to X$
, a partition element
$P\in {\boldsymbol P}(f)$
is called wandering if there exists
$M\in {\mathbb N}$
such that
$f^n(P)\cap P=\emptyset $
for all
$n>M$
. The set
${\mathbb W}(f)\subset {\boldsymbol P}(f)$
, consisting of all wandering partition elements, is called the wandering elements set. We set
$W(f)=\bigcup _{P\in {\mathbb W}(f)}P$
. (Note that this set is not the usual wandering set, which is much bigger.)
Definition 2.13. (Non-wandering set)
The complement
$N{\mathbb W}(f)$
of the wandering elements set is called the non-wandering elements set: that is,
$N{\mathbb W}(f)= \{P\in {\boldsymbol P}(f): P\not \in {\mathbb W}(f)\}$
. We set
$NW(f)=\bigcup _{P\in N{\mathbb W}(f)}P$
.
Note that, for a Markov map, the set
$N{\mathbb W}(f)$
corresponds to the non-wandering set of the dynamical system 
defined in Remark 2.9. Accordingly, our definition of
$NW(f)$
should not be confused with the usual non-wandering set of f, which, for a Markov map, consists of finitely many points (the set of periodic points; see Theorem 2.8). Hereby, we state our definition of topological stability.
Definition 2.14. (Topological stability)
Let
$\mathfrak {C}$
be contained in the set of piecewise contractions from
$X\to X$
and let
$d $
be a metric defined on the set of piecewise contractions. We say that
$(\mathfrak {C},d)$
is topologically stable if, for every
$f\in \mathfrak {C}$
, there exists a
$\delta>0$
such that, for any piecewise contraction
$g\in {\mathfrak C}$
with
$d(f,g)<\delta $
, f is semi-conjugate to g and g is semi-conjugate to f on the respective non-wandering sets: that is, there exist continuous functions
$H_1: {NW}(f)\to {NW}(g),\ H_2:{NW}(g)\to {NW}(f)$
such that
$H_1\circ f= g\circ H_1$
,
$f\circ H_2= H_2 \circ g$
and
$H_1(NW(f))=NW(g)$
,
$H_2(NW(g))=NW(f)$
.
This definition of topological stability is somewhat different from the standard definition (see [[Reference Katok and Hasselblatt18], Definition 2.3.5]). In general, topological stability is defined for homeomorphisms such that one of them is semi-conjugate to the other (whereas we have semi-conjugacies for both sides), but on the entire space. Here it is necessary to define it only on the non-wandering set which, in fact, is the only subset of the space that contributes to the long-time dynamics. We can interpret our definition as stating that the long-time dynamics of two such functions are qualitatively the same.
Theorem 2.15. The set of Markov piecewise injective contractions (as defined in Definitions 2.7, 2.10) is topologically stable in the
$d_1$
topology.
The proof of the above theorem is given in §3. To state our result on density, we restrict ourselves to piecewise smooth contractions, which are defined as follows.
Definition 2.16. (Piecewise smooth contraction)
A piecewise injective contraction f with partition
${\boldsymbol P}(f)=\{P_i\}, i\in \{1,2,\ldots ,m\}$
and injective extensions
$\tilde f_i:U_i\to {\mathbb R}^d$
is called a piecewise smooth contraction if, for every
$i\in \{1,2,\ldots ,m\}$
:
-
•
$\|\tilde f_i\|_{{\mathcal C}^3}<\infty $
; -
•
$\|\tilde f_i^{-1}\|_{{\mathcal C}^1}<\infty $
; and -
•
$\partial P_i$
is contained in the union of finitely many
${\mathcal C}^2 (d-1)$
-dimensional manifolds
$\{M_j\}$
,
$\partial M_j\cap \overline P_i=\emptyset $
. Such manifolds are pairwise transversal, and the intersection of any set of such manifolds consists of a finite collection of
${\mathcal C}^2$
manifolds.
For a piecewise smooth contraction f, we define the extended-metric
$$ \begin{align*} d_2(f,g)=\begin{cases} \sup\limits_{P\in{\boldsymbol P}(f)}\|f-g\|_{{\mathcal C}^2(P)} &\text{if } {\boldsymbol P}(f)={\boldsymbol P}(g),\\ \ \ \infty & \text{otherwise}. \end{cases} \end{align*} $$
Note that
$d_2(f,g)\geq \rho (f,g)$
. The following density result is proved in §4.
Theorem 2.17. Piecewise smooth Markov contractions are
$d_2$
-dense in the space of piecewise smooth contractions.
Remark 2.18. Theorems 2.8, 2.11 and 2.15 show that, for a piecewise contraction, to be Markov means being stable under a rather weak topology (
$d_1$
). Instead, Theorem 2.17 shows that to be Markov means being dense under a quite strong topology (
$d_2$
). These theorems collectively show that, for a piecewise contraction, being Markov is generic; hence, to have a Cantor set as an attractor is rare.
As already mentioned, a result on density is present in the literature (see [Reference Catsigeras, Guiraud, Meyroneinc and Ugalde8]). However, it is proved under a much coarser metric as compared with
$d_2$
. More importantly, it assumes that the maps have the separation property, which implies that they are globally injective, whereas we assume only piecewise injectivity.
3. Openness and topological stability
Recall that, for any piecewise contraction f,
${\boldsymbol P}(f)$
stands for the maximal partition, so
$\partial {\boldsymbol P}(f)=\Delta (f)$
. In addition, for any
$n>1$
, the elements of partition
${\boldsymbol P}(f^n)$
are given by equation (2.1), and
$\#{\boldsymbol P}(f^n)=m_n$
,
$m_1=m$
.
Proof of Theorem 2.8
Let f satisfy
$\Lambda (f)\cap \Delta (f)=\emptyset $
. For all
$n\in {\mathbb N}$
,
$\overline {f^{n+1}(X)}\subset \overline {f^{n}(f(X))}\subset \overline {f^n(X)}$
implies that
$\{\overline {f^n(X)}\}$
is a nested sequence of non-empty compact sets. Further, Cantor’s intersection theorem implies that
$\Lambda (f)=\bigcap _{n\in {\mathbb N}} \overline {f^n(X)}\neq \emptyset $
and it is closed. Accordingly, there exists
$\varepsilon>0$
such that
$d_0(\Lambda (f),\Delta (f))>\varepsilon $
. We claim that, for any
$\varepsilon>0$
, there exists
$N_{\varepsilon }\in {\mathbb N}$
such that, for
$n\geq N_{\varepsilon }$
,
$\overline {f^n(X)}\subset B_{\varepsilon /2}(\Lambda (f))$
. (For
$r>0$
and a set A,
$ B_r(A)=\{y: \text { there exists } x\in A\ \mathrm {such\ that}\ d_0(x,y)<r $
}.) Indeed, if this was not the case, then there would exist a sequence
$\{n_j\}$
,
$n_j\to \infty $
, such that
$\overline {f^{n_j}(X)}\cap B_{\varepsilon /2}(\Lambda (f))^c\neq \emptyset $
. It follows that, for each
$n\in {\mathbb N}$
, there exists j such that
$n_j\geq n$
, and hence
$$ \begin{align*} \overline{f^{n}(X)}\cap B_{\varepsilon/2}(\Lambda(f))^c\supset \overline{f^{n_j}(X)}\cap B_{\varepsilon/2}(\Lambda(f))^c\neq \emptyset, \end{align*} $$
which, taking the intersection on n, yields a contradiction. Consequently, for every
${P\in {\boldsymbol P}(f^{N_\varepsilon })}$
, there exists
$Q \in {\boldsymbol P}(f^{N_{\varepsilon }})$
such that
$\overline {f^{N_\varepsilon }(P)}\subset Q$
; otherwise, there would exist
$x\in f^{N_\varepsilon }(P)\cap \partial {\boldsymbol P}(f^{N_{\varepsilon }})$
, that is, a
$k\in {\mathbb N}$
such that
$$ \begin{align*} f^{k}(x)\in f^{k+N_{\varepsilon}}(P)\cap \partial {\boldsymbol P}(f)\subset \overline{f^{k+N_\varepsilon}(X)}\cap\Delta(f)\subset B_{\varepsilon/2}(\Lambda(f))\cap\Delta(f)=\emptyset, \end{align*} $$
which is a contradiction.
Conversely, let f be a piecewise Markov contraction with stabilization time
$N\in {\mathbb N}$
. By definition, for every
$P\in {\boldsymbol P}(f^N)$
, there exists
$Q\in {\boldsymbol P}(f^N)$
such that
$\overline {f^N(P)}\subset Q$
: that is,
$\overline {f^N(P)}\cap \partial {\boldsymbol P}(f^N)=\emptyset $
. Note that
$\overline {f(X)}=\bigcup _{i=1}^m\overline {f(\tilde P_i)}\subset \bigcup _{i=1}^m \overline {f(P_i)} $
, where
$\tilde P_i$
is as given in Definition 2.1. Thus,
$$ \begin{align*} \Lambda(f)=\bigcap_{n\in{\mathbb N}}\overline{f^n(X)}\subset \bigcap_{n\in{\mathbb N}}\bigcup_{P\in{\boldsymbol P}(f^n)}\overline{f^n(P)}\subset \bigcup_{P\in{\boldsymbol P}(f^N)}\overline{f^N(P)}\subset \bigcup_{Q\in {\boldsymbol P}(f^N)}Q \end{align*} $$
implies that
$\Lambda (f)\cap \partial {\boldsymbol P}(f)=\emptyset $
.
To prove the second part of the theorem, let
$x\in \Lambda (f)$
and let N be the stabilization time. By definition, for each
$k\in {\mathbb N}$
, there exists
$y_k\in Q_{k}\in {\boldsymbol P}(f^N)$
such that
$x=f^{kN}(y_k)$
. In addition, there exists
$P\in {\boldsymbol P}(f^N)$
such that
$x\in P$
. This implies that
${\overline {f^{kN}(Q_k)}}\subset P$
for all
$k\in {\mathbb N}$
. Let
$l:=\# {\boldsymbol P}(f^N)$
. (For a discrete set
$M,\ \# M$
denotes the cardinality of M.) Then there exists
$k_1\in \{0,\ldots , l\}$
such that
$P=Q_{k_1}$
and hence
$\overline {f^{k_1N}(P)}\subset P$
. By the contraction mapping theorem,
$f^{k_1N}:\overline {P}\to \overline {P}$
has a unique fixed point, say,
$z\in \overline {P}$
. Let
$j\in {\mathbb N}$
be the smallest integer for which
$f^j(z)=z$
. Then
$x\in \bigcap _{n\in {\mathbb N}}\overline { f^{nj}(P)}=\{z\}$
. Hence,
$\Lambda (f)$
consists of periodic orbits.
To prove the result on openness and topological stability, we first prove that the functions
$\rho ,\ d_1$
, defined in §2, are, in fact, metrics on the set of piecewise contractions. Note that, by definition of
$H({\boldsymbol P}(f),{\boldsymbol P}(g))$
, if
$\# {\boldsymbol P}(f)\neq \# {\boldsymbol P}(g)$
, then
$H({\boldsymbol P}(f),{\boldsymbol P}(g))=\emptyset $
.
Lemma 3.1.
$\rho $
is a metric.
Proof. Let
$f,g,h$
be piecewise contractions.
If
$\rho (f,g)=0$
, then there exists a sequence
$\psi _n\in H({\boldsymbol P}(f),{\boldsymbol P}(g))$
,
$\|\psi _n-\mathrm {id}\|_{{\mathcal C}^0(X,X)} \to 0$
and
$\|f-g\circ \psi _n\|_{{\mathcal C}^0(X,X)}\to 0$
as
$n\to \infty $
, which implies that
$\psi _n\to \mathrm {id}$
as
$n\to \infty $
which further implies that
${\boldsymbol P}(f)\hspace{-1pt}=\hspace{-1pt}{\boldsymbol P}(g)$
. Also,
$\psi _n\hspace{-1pt}\to\hspace{-1pt} \mathrm {id}$
means that
$f\hspace{-1pt}-\hspace{-1pt}g\circ \psi _n\hspace{-1pt}\to\hspace{-1pt} f\hspace{-1pt}-\hspace{-1pt}g$
, and hence
$f=g$
.
Next, we check the symmetry of
$\rho $
. If
$H({\boldsymbol P}(f),{\boldsymbol P}(g))\kern1.3pt{=}\kern1.3pt\emptyset $
, then
${\rho (f,g)\kern1.3pt{=}\kern1.3pt\rho (g,f)\kern1.3pt{=}\kern1.3pt A}$
. If
$H({\boldsymbol P}(f),{\boldsymbol P}(g))\neq \emptyset $
, then
$$ \begin{align*} \rho(f,g)&=\inf\limits_{\psi\in H({\boldsymbol P}(f),{\boldsymbol P}(g))} \{\|\psi-\mathrm{id} \|_{{\mathcal C}^0(X,X)}+\|f-g\circ\psi\|_{{\mathcal C}^0(X,X)}\}\\ & = \inf\limits_{\psi^{-1}\in H({\boldsymbol P}(g),{\boldsymbol P}(f))} \{\|\psi^{-1}-\mathrm{id}\|_{{\mathcal C}^0(X,X)}+ \|f\circ\psi^{-1}-g\|_{{\mathcal C}^0(X,X)}\}\\ &=\rho(g,f). \end{align*} $$
It remains to check the triangle inequality.
To show that
$\rho (f,g)\leq \rho (f,h)+\rho (g,h)$
, consider the following cases.
If
$H({\boldsymbol P}(f),{\boldsymbol P}(g))=\emptyset $
, then
$\rho (f,g)=A$
, and
$H({\boldsymbol P}(f),{\boldsymbol P}(h))=\emptyset $
or/and
$H({\boldsymbol P}(g), {\boldsymbol P}(h))=\emptyset $
: that is, either one of the two or both are empty sets. Therefore,
$\rho (f,h)=A$
or
$\rho (g,h)=A$
and so
$\rho (f,g)=A \leq \rho (f,h)+\rho (h,g)$
.
If
$H({\boldsymbol P}(f),{\boldsymbol P}(g))\neq \emptyset $
, then there are the following two possibilities.
-
(1) If
$H({\boldsymbol P}(f),{\boldsymbol P}(h))=\emptyset $
and
$H({\boldsymbol P}(h), {\boldsymbol P}(g))=\emptyset $
, then
$\rho (f,h)+\rho (g,h)\geq A$
and
$$ \begin{align*} \hspace{1.1cm}\rho(f,g)& =\inf\limits_{\psi\in H(P(f),P(g))} \{\|\psi-\mathrm{id}\|_{{\mathcal C}^0(X,X)}+\|f-g\circ\psi\|_{{\mathcal C}^0(X,X)}\}\\ &\leq2 \operatorname{diam}(X). \end{align*} $$
Since
$A\geq 2 \operatorname {diam}(X)$
, we have the result. -
(2)
$H({\boldsymbol P}(f),{\boldsymbol P}(g))\neq \emptyset $
, and
$H({\boldsymbol P}(g),{\boldsymbol P}(h))\neq \emptyset $
.Given
$\phi \in H({\boldsymbol P}(g),{\boldsymbol P}(h))$
and
$\varphi \in H({\boldsymbol P}(f),{\boldsymbol P}(h))$
, the homeomorphism
${\psi =\phi ^{-1}\circ \varphi \in H({\boldsymbol P}(f), {\boldsymbol P}(g))}$
, and hence
$$ \begin{align*} \rho(f,g)&=\inf\limits_{\psi\in H({\boldsymbol P}(f), {\boldsymbol P}(g))}\{\|\psi-\mathrm{id}\|_{{\mathcal C}^0(X,X)}+ \|f-g\circ\psi\|_{{\mathcal C}^0(X,X)}\}\\[2pt] &\leq \inf\limits_{\varphi\in H({\boldsymbol P}(f),{\boldsymbol P}(h))}\inf\limits_{\phi\in H({\boldsymbol P}(g),{\boldsymbol P}(h))} \{\|\phi^{-1}\circ\varphi-\mathrm{id}\|_{{\mathcal C}^0(X,X)}\\[2pt] &\quad +\|f-h\circ\varphi\|_{{\mathcal C}^0(X,X)} +\|h\circ\varphi-g\circ\phi^{-1}\circ\varphi\|_{{\mathcal C}^0(X,X)}\} \\[2pt] &\leq\! \inf\limits_{\varphi\in H({\boldsymbol P}(f),{\boldsymbol P}(h))}\inf\limits_{\phi\in H({\boldsymbol P}(g),{\boldsymbol P}(h))} \{\|\phi^{-1}-\mathrm{id}\|_{{\mathcal C}^0(X,X)}+\|f-h\circ\varphi\|_{{\mathcal C}^0(X,X)}\\[2pt] &\quad +\|\varphi-\mathrm{id}\|_{{\mathcal C}^0(X,X)}+\|h\circ\varphi-g\circ\phi^{-1}\circ\varphi\|_{{\mathcal C}^0(X,X)}\} \\[2pt] &=\inf\limits_{\phi\in H({\boldsymbol P}(g),{\boldsymbol P}(h))} \{\|\phi^{-1}-\mathrm{id}\|_{{\mathcal C}^0(X,X)}+\|h\circ\varphi-g\circ\phi^{-1}\circ\varphi\|_{{\mathcal C}^0(X,X)}\}\\[2pt] &\quad + \inf\limits_{\varphi\in H({\boldsymbol P}(f),{\boldsymbol P}(h))}\{\|\varphi-\mathrm{id}\|_{{\mathcal C}^0(X,X)}+\|f-h\circ\varphi\|_{{\mathcal C}^0(X,X)}\}\\[2pt] &=\rho(f,h)+\rho(g,h). \end{align*} $$
Hence,
$\rho (\cdot ,\cdot )$
is a metric.
Lemma 3.1 implies that
$d_1$
is also a metric since the series is convergent.
Proof of Theorem 2.11
Let
$f $
be Markov. We want to prove that there exists a neighbourhood of f consisting of only Markov contractions. Since f is Markov, there exists
$N\in {\mathbb N}$
such that the maximal partition
${\boldsymbol P}(f)$
of f stabilizes with stabilization time N. Let
$g\neq f$
be a piecewise contraction such that
$d_1(f,g)<\delta $
for
$0<\delta < \sigma ^N A$
(recall that
$A=\operatorname {diam}(X)$
).
We show that the partition of g stabilizes with the same stabilization time N.
Note that, for all
$n\leq N$
,
$\rho (f^n,g^n)<A$
which implies that
$H({\boldsymbol P}(f^n),{\boldsymbol P}(g^n))\neq \emptyset $
. Let
$P\in {\boldsymbol P}(g^N)$
. Then there exist
$\psi _N\in H({\boldsymbol P}(f^N),{\boldsymbol P}(g^N))$
and
$P^\prime \in {\boldsymbol P}(f^N)$
such that
$\psi _N(P^\prime )~=~P$
. By stabilization, for
$P^{\prime }$
, there exists a unique
$Q^\prime \in {\boldsymbol P}(f^N)$
such that
$\overline {f^N(P^\prime )}\subset Q^\prime $
. In addition,
$\psi _N(Q^\prime )=Q$
for some
$Q\in {\boldsymbol P}(g^N)$
. Now,
$\overline {f^N(P^\prime )}\subset Q^\prime $
and
$Q^\prime $
being open implies that
$\varepsilon =\min _{P^\prime \in {\boldsymbol P}(f^N)}d_0(\overline {f^N(P^\prime )}, \partial Q^\prime )>0$
. Choosing
$0<\delta <\varepsilon \sigma ^N/3$
, we claim that
$\overline {g^N(P)}\subset Q$
. Indeed, if there exists
$x\in \overline {g^N(P)}\cap \partial Q$
, then there exists a sequence
$\{x_k\}\in g^N(P)\cap Q$
such that
$x_k\to x$
as
$k \to \infty $
. Let
$y_k\in P$
such that
$g^N(y_k)=x_k$
. Note that
$\psi _N^{-1}(x)\in \partial Q^\prime $
. Now,
$$ \begin{align*} \varepsilon&<d_0(f^N(\psi_N^{-1}(y_k)),\psi_N^{-1}(x))\\[2pt] &\leq d_0(f^N(\psi_N^{-1}(y_k)),g^N(y_k))+d_0(g^N(y_k),x)+d_0(x,\psi_N^{-1}(x))\\[2pt] &\leq \|f^N\circ \psi_N^{-1}-g^N\|_{{\mathcal C}^0(X,X)}+\|\mathrm{id}-\psi_N^{-1}\|_{{\mathcal C}^0(X,X)}+ d_0(x_k,x)\\[2pt] &=\rho(f^N,g^N)\!+\!d_0(x_k,x)(\text{taking infimum over } \psi_N\in H({\boldsymbol P}(f^N),{\boldsymbol P}(g^N)) \text{ on both sides})\\[2pt] &<\delta \sigma^{-N}+ d_0(x_k,x) \to \varepsilon\sigma^N\sigma^{-N}/3 \text{ as } k\to \infty, \end{align*} $$
and we get
$\varepsilon <\varepsilon /3$
, which is a contradiction. Hence, if f is Markov with N as the stabilization time, then, for
$$ \begin{align*} \delta<\min\bigg\{\sigma^N A, \frac{\sigma^N}{3}\inf_{P,Q\in{\boldsymbol P}(f^N)}d_0(\overline{f^N(P)},\partial Q)\bigg\}, \end{align*} $$
all piecewise contractions g, with
$d_1(f,g)<\delta $
, are Markov contractions and the stabilization time of g is also N. Thus, the collection of Markov contractions is open.
To prove Theorem 2.15, we first need to prove the following lemma which, in itself, brings some important information about the dynamics of a Markov contraction.
Lemma 3.2. Let f be a Markov contraction with maximal partition
${\boldsymbol P}(f)$
and stabilization time N. Then, for every
$P\in {\boldsymbol P}(f^N)$
, there exists
$Q\in {\boldsymbol P}(f^N)$
such that
$f(P)\subset Q$
.
Proof. By Definition 2.6, there exists
$P^\prime \in {\boldsymbol P}(f^N)$
such that
$f^N(P)\subset P^\prime $
. By the definition given in equation (2.1), there exist partition elements
$\{P_i\}_{i=0}^{N-1}$
(not necessarily distinct) in
${\boldsymbol P}(f)$
such that
$$ \begin{align*} P= P_0\cap f^{-1}P_1\cap f^{-2}P_2\cap\cdots\cap f^{-(N-1)}P_{N-1}. \end{align*} $$
Similarly, there exist partition elements
$\{P_j^\prime \}_{j=1}^{N-1}$
in
${\boldsymbol P}(f)$
such that
$$ \begin{align*} P^\prime = P_0^\prime \cap f^{-1}P_1^\prime \cap f^{-2}P_2^\prime\cap\cdots\cap f^{-(N-1)}P_{N-1}^\prime. \end{align*} $$
Let
$x\in P$
. Then
$f^k(f(x))\in P_{k+1}$
for every
$k\in \{0,1,\ldots ,N-2\}$
and
$f^N(x)= f^{N-1}f(x)\in P^\prime $
, which implies that
$f(x)\in f^{-(N-1)}(P^\prime _0)$
. Consequently,
$$ \begin{align*} f(x)\in P_1\cap f^{-1}P_2\cap\cdots\cap f^{-(N-2)}P_{N-1}\cap f^{-(N-1)}P^\prime_0=Q\in {\boldsymbol P}(f^N). \end{align*} $$
Since
$x\in P$
is arbitrary,
$f(P)\subset Q\in {\boldsymbol P}(f^N)$
.
Finally, to prove Theorem 2.15, we restrict to piecewise injective contractions and prove the stability result under the metric
$d_1$
. Recall that, for every piecewise injective contraction f, with maximal partition
${\boldsymbol P}(f)=\{P_1,\ldots ,P_m\}$
, there exists
$U_i=\mathring U_i\supset \overline {P_i}$
and an injective continuous extension
$\tilde {f}:U_i\to {\mathbb R}^d$
such that
$\tilde f|_{P_i}=f|_{P_i}$
. In this proof, we always consider these extensions which, to alleviate notation, we still denote by f.
Our strategy for the following proof is as follows. For piecewise injective Markov contractions f and g with maximal partitions
${\boldsymbol P}(f)$
and
$ {\boldsymbol P}(g)$
, respectively, and for some
$\delta>0$
,
$d_1(f,g)<\delta $
, we construct semi-conjugacies from
$NW(f)$
, the non-wandering set for f, to
$NW(g)$
, the non-wandering set for g, using a homeomorphism between the partitions given in the definition of the metric
$\rho $
.
Proof of Theorem 2.15
Let
$N\in {\mathbb N}$
be the stabilization time of
${\boldsymbol P}(f)$
. By Theorem 2.11, there exists
$\delta>0$
such that, if
$d_1(f,g)<\delta $
, then
${\boldsymbol P}(g)$
also stabilizes at time N. By Theorem 2.8, the attractors of f and g consist of eventually periodic orbits. Let
$P\in {\boldsymbol P}(f^N)$
be a periodic element of the partition. Then there exists
$n_0\in {\mathbb N}$
such that
$\overline {f^{n_0}(P)}\subset P$
. Then
$d_1(f,g)<\delta $
implies that there exists
$Q\in {\boldsymbol P}(g^N)$
such that
$\overline {g^{n_0}(Q)}\subset Q$
. By the contraction mapping theorem, for
$f^{n_0}: \overline {P}\to \overline {P}, g^{n_0}: \overline {Q}\to \overline {Q}$
, there exist
$x_f$
and
$x_g$
, the unique fixed points of
$f^{n_0}$
and
$g^{n_0}$
, respectively, in P and Q.
Using Lemma 3.2 inductively, let
$P_i,\ Q_i$
be the partition elements in
${\boldsymbol P}(f^N), {\boldsymbol P}(g^N)$
, respectively, such that
$ f^i(\overline P)\subset P_i $
,
$ g^i(\overline Q)\subset Q_i$
. Let
$\widehat {P}= \bigcup _{i=1}^{n_0} {\overline P_i}$
,
$ \widehat {Q}=\bigcup _{i=1}^{n_0}{\overline Q_i}$
. Then, for each
$i\in {\mathbb N}$
,
$\widehat {P}_i=f^i(\widehat {P})\subset \widehat {P}$
,
$\widehat {Q}_i=g^i(\widehat {Q})\subset \widehat {Q} $
.
Note that, for
$\delta $
small enough, we have
$ H({\boldsymbol P}(f^n),{\boldsymbol P}(g^n))\neq \emptyset $
for every
$n\leq n_0$
: that is, there exists
$\psi \in H({\boldsymbol P}(f^{n_0}),{\boldsymbol P}(g^{n_0}))$
such that
$\|\psi -\mathrm {id}\|_{{\mathcal C}^0(X,X)}<\delta $
and
$\|f^n-g^n\circ \psi \|_{{\mathcal C}^0(X,X)}<\delta $
, and thus
$\psi (\widehat {P})=\widehat {Q}$
.
Next, define
$\widehat g=\psi \circ f\circ \psi ^{-1}$
. Since, by Definition 2.10,
$f^{-1}$
is well defined on
$\widehat P$
, we have that
$\widehat g$
is invertible on
$\widehat Q$
and
$\widehat g^{-1}(\partial \widehat Q)=\psi \circ f^{-1}( \partial \widehat P)$
.
Next, for
$\varepsilon>0$
small enough, let
$\widehat P_{1, \varepsilon }$
be the
$\varepsilon $
-neighbourhood of
$f(\widehat P)$
and let
$\widehat P_{c,\varepsilon }$
be the
$\varepsilon $
-neighbourhood of
${\widehat P}^\complement $
, the complement of
$\widehat P$
. Similarly, let
$\widehat Q_{c,\varepsilon }=\psi \circ f^{-1}(\widehat P_{c,\varepsilon })$
and
$\widehat Q_{1,\varepsilon }=\psi \circ f^{-1}(\widehat P_{1,\varepsilon })$
. By the Markov property, we have
$\overline {\widehat P}_{1,\varepsilon }\cap \overline {\widehat P}_{c,\varepsilon }=\emptyset $
, and thus
$\overline {\widehat Q}_{1,\varepsilon }\cap \overline {\widehat Q}_{c,\varepsilon }=\emptyset $
. Hence, by Urysohn’s lemma, there exists a function
$\theta \in {\mathcal C}^0(X,[0,1])$
such that
$\theta |_{\overline {\widehat Q}_{1, \varepsilon }}=1$
and
$\theta |_{\overline {\widehat Q}_{c,\varepsilon }}=0$
. Finally, define the continuous functions
$$ \begin{align*} \begin{split} &\tilde g(x)=\theta(x) g(x)+(1-\theta(x))\widehat g(x),\\ &h_0(x)=\tilde g\circ \psi\circ f^{-1}(x)\quad\text{for all } x\in \overline{\widehat P\setminus\widehat P_1}. \end{split} \end{align*} $$
Lemma 3.3. Provided
$\delta>0$
is small enough, we have
$h_0(\widehat P\setminus \widehat P_1)=\widehat Q\setminus \widehat Q_1$
,
$h_0(\partial \widehat P)=\partial \widehat Q$
,
$h_0(\partial \widehat P_1)=\partial \widehat Q_1$
.
Proof. First, using the properties of the homeomorphism
$\psi $
, we have, for each
$x\kern1.2pt{\in}\kern1.2pt \overline {\widehat P\setminus \widehat P_1}$
,
$$ \begin{align} |h_0(x)-x|=| \theta(x) [g\circ \psi\circ f^{-1}(x)-x]+(1-\theta(x))[\psi(x)-x]|\leq 2\delta. \end{align} $$
If
$x\in \widehat P_{c,\varepsilon }\cap \widehat P$
, then
$\psi \circ f^{-1}(x)\in \widehat Q_{c,\varepsilon }$
, and thus
$h_0(x)=\widehat g\circ \psi \circ f^{-1}(x)=\psi (x)$
. Whereas, if
$x\in \widehat P_{1,\varepsilon }$
, then
$\psi \circ f^{-1}(x)\in \widehat Q_{1,\varepsilon }$
, and thus
$h_0(x)=g\circ \psi \circ f^{-1}(x)$
. In addition,
$$ \begin{align*} \begin{split} &h_0(\partial \widehat P)=\psi(\partial \widehat P)=\partial \widehat Q,\\ &h_0(\partial \widehat P_1)=g\circ \psi\circ f^{-1}(\partial\widehat P_1)=g\circ \psi(\partial\widehat P)=g(\partial \widehat Q)=\partial \widehat Q_1. \end{split} \end{align*} $$
Also, if
$\delta $
is small enough, then
$h_0$
is invertible on
$(\widehat P_{c,\varepsilon }\cup \widehat P_{1, \varepsilon })\cap \widehat P$
. Thus, to prove surjectivity, it suffices to prove that each
$p\in (\widehat Q\setminus \widehat Q_1)\setminus h_0((\widehat P_{c,\varepsilon }\cup \widehat P_{1, \varepsilon })\cap \widehat P)$
belongs to
$h_0(\widehat P\setminus \widehat P_1)$
. Let
$B=\{z\in {\mathbb R}^{n}\;:\; \|z\|\leq 3\delta \}$
,
$x=p+z$
and
$h_0(x-p)=x-h_0(x)$
. Then
$h_0(x)=p$
is equivalent to
$$ \begin{align*} z=h_0(z). \end{align*} $$
Since equation (3.1) implies that
$h_0(B)\subset B$
, by Brouwer’s fixed-point theorem it follows that there exists at least one
$z\in B$
such that
$h_0(z+p)=p$
and, for
$\delta \leq \varepsilon /6$
,
$z+p \in \widehat P\setminus \widehat P_1$
.
For the sake of convenience, let
$\widehat P_0=\widehat P$
and
$\widehat Q_0=\widehat Q$
. For every
$i\in {\mathbb N}$
, we define
${h_{i}:\overline {\widehat {P}_i\backslash \widehat {P}_{i+1}}\to \overline {\widehat {Q}_i\backslash \widehat {Q}_{i+1}}}$
as
$h_{i}(x)=g\circ h_{i-1}\circ f^{-1}(x)$
. Thus, we define the semi-conjugacy
$H:\overline {\widehat {P}}\to \overline {\widehat {Q}}$
as
$$ \begin{align*} H(x)=\begin{cases} h_{i}(x), & x\in \overline{\widehat{P}_i\backslash \widehat{P}_{i+1}},\\ x_g, & x=x_f. \end{cases} \end{align*} $$
H is continuous because, for
$x\in \partial \widehat {P}_i, H(x)=h_{i+1}(x)=g\circ h_i\circ f^{-1}(x)=h_i(x)$
and, for any sequence
$(x_i)\in \widehat {P}_i\backslash \widehat {P}_{i+1}$
with
$x_i\to x_f$
as
$i\to \infty $
,
$(Hx_i)\in \widehat {Q}_i\backslash \widehat {Q}_{i+1}$
, so
$H(x_i) \to x_g=H(x_f)$
. Indeed, H is surjective, for
$p\in \overline {\widehat {Q}}$
, and there exists
$i\in {\mathbb N}\cup \{0\}$
such that
$p\in \overline {\widehat Q_i\setminus \widehat Q_{i+1}}$
. We use induction on i. If
$i=0$
, then, using Lemma 3.3, there exists
$z\in \overline {\widehat Q_i\setminus \widehat Q_{i+1}}$
such that
$h_0(z)=p$
. Instead, if
$i\neq 0$
, then
$p_1=g^{-1}(p)\in \overline {\widehat Q_{i-1}\setminus \widehat Q_{i}}$
. Inductively, as
$h_{i-1}$
is surjective, there exists
$q_1\in \overline {\widehat P_{i-1}\setminus \widehat P_{i}}$
such that
$h_{i-1}(q_1)=p_1$
. Now, by definition,
$q=f(q_1)\in \overline {\widehat P_{i}\setminus \widehat P_{i+1}}$
, and finally
${g\circ h_i\circ f^{-1}(q)=p}$
.
Furthermore, H is the wanted semi-conjugacy between
$f,g$
, on
$\widehat {P}$
and
$\widehat {Q}$
because, for
$x\in \widehat {P}$
,
$$ \begin{align*} H_{\widehat{P}}\circ f(x)= h_{i+1}\circ f(x)=g\circ h_i\circ f^{-1}\circ f(x)=g\circ h_i(x) =g\circ H(x). \end{align*} $$
To obtain a semi-conjugacy on the whole of
$NW(f)$
, we repeat the same steps for every periodic partition element
$P\in {\boldsymbol P}(f^N), Q\kern1.2pt{\in}\kern1.2pt {\boldsymbol P}(g^N)$
with the same
${\psi \kern1.2pt{\in}\kern1.2pt H({\boldsymbol P}(f^N),{\boldsymbol P}(g^N))}$
. Calling
$\{\widehat {P}^k\}$
the collection of the union of elements associated to a periodic orbit, we have
$\bigcup _k \widehat {P}^k=NW(f)$
. Pasting these functions together, we obtain a function
$\Theta :{NW}(f)\to {NW}(g)$
with
$\Theta (x)=H_{\widehat {P}^k}(x)$
for
$ x\in \widehat {P}^k\subset NW(f)$
. Then
$\Theta $
is continuous on
$NW(f)$
as
$\Theta |_{\partial \widehat {P}^k}=\psi |_{\partial \widehat {P}^k}$
for every k.
To construct the semi-conjugacy from the other side, we use the fact that
$\psi $
is a homeomorphism and repeat the same construction using
$\psi ^{-1}$
instead of
$\psi $
and switching the roles of f and g.
4. Density
We prove Theorem 2.17 in two steps. First, we show that Markov maps are dense in a special class of systems (piecewise strongly contracting) and then we will show that such a class is itself dense in the collection of piecewise smooth contractions.
Definition 4.1. (Piecewise strongly contracting)
A piecewise smooth contraction f with contraction coefficient
$\unicode{x3bb} $
and maximal partition
${\boldsymbol P}(f)=\{P_1,P_2,\ldots ,P_m\}$
is said to be piecewise strongly contracting if there exists
$p\in {\mathbb N}$
such that
$\unicode{x3bb} ^pm_p<1/2$
, where
${m_p=\#{\boldsymbol P}(f^p)}$
.
We prove the following results.
Proposition 4.2. Markov maps are
$d_2$
-dense in the collection of piecewise strong contractions.
Proposition 4.3. Piecewise strong contractions are
$d_2$
-dense in the collection of piecewise smooth contractions.
These two propositions readily imply our main result.
Proof of Theorem 2.17
Let f be a piecewise smooth contraction. By Proposition 4.3, for each
$\varepsilon>0$
, there exists a piecewise strong contraction
$f_1$
such that
$ d_2(f,f_1)<\varepsilon /2$
. In addition, by Proposition 4.2, there exists a piecewise smooth Markov contraction
$f_2$
such that
$ d_2(f_1,f_2)<\varepsilon /2$
, and hence the result.
In the rest of the paper, we prove Propositions 4.2 and 4.3.
The basic idea of the proof is to introduce iterated function systems (IFSs) associated with the map. The attractor of the IFS is greater than the one of the map (see §4.1 for the relationship between the two sets), and hence if we can prove that the attraction of the IFS is disjoint from the discontinuities of the map, so will be the attractor of the map. The advantage is that, in this way, the study of the boundaries of the elements of
${\boldsymbol P}(f^n)$
is reduced to the study of the pre-images of the discontinuities of f under the IFS. Hence, we can iterate smooth maps rather than discontinuous ones.
This advantage is first exploited in §4.2, where we prove Proposition 4.2 using an argument that is, essentially, a quantitative version of Sard’s theorem.
To prove Proposition 4.3, the rough idea is to use a transversality theorem (see Appendix B) to show that if a lot of pre-images intersect, then, generically, their intersection should have smaller and smaller dimensions until no further intersection is generically possible. Unfortunately, if we apply a transversality theorem to a composition of maps of the IFS, we get a perturbation of the composition and not of the single maps. How to perturb the single maps in such a way that the composition has the wanted properties is not obvious.
Our solution to this problem is to make sure that if we perturb the maps in a small neighbourhood B, and we consider arbitrary compositions of the perturbed maps, then all the images of B along the composition never intersect B. Hence, if we restrict the composition to B, all the maps, except the first, will behave as their unperturbed version. To ensure this, it suffices to prove that such compositions have no fixed points near the singularity manifolds (such an implication is proved in Lemma 4.10). To this end, in Propositions 4.11 and 4.17, we show that one can control the location of the fixed points of the compositions of the map of the IFS by an arbitrarily small perturbation.
After this, we can finally set up an inductive scheme to ensure that the pre-images of the discontinuity manifolds keep intersecting transversally. This is the content of Proposition 4.19 from which Proposition 4.3 readily follows.
4.1. IFSs associated to the map and their properties
We start by recalling the definition of an IFS relevant to our argument and exploring some of its properties.
Definition 4.4. (IFS)
The set
$\Phi =\{\phi _1, \phi _2, \ldots ,\phi _m\}, m\geq 2$
, is an IFS if each map
${\phi _i:{\mathbb R}^d\to {\mathbb R}^d}$
is a Lipschitz contraction. (In the following, we will consider only
${\mathcal C}^3$
maps
$\phi _i$
.)
Let f be a piecewise smooth contraction with maximal partition
${\boldsymbol P}(f)$
. By analogy with [Reference Nogueira, Pires and Rosales23], we define an IFS associated to f as follows.
By Definition 2.16,
$\tilde f|_{U_i}$
is
${\mathcal C}^3$
for every i and
$D\tilde f|_{U_i}\leq \unicode{x3bb} <1$
. Using the
${\mathcal C}^r$
version of Kirszbraun–Valentine theorem A.2, we obtain a
${\mathcal C}^3$
extension
$\phi _i:{\mathbb R}^d\to {\mathbb R}^d$
of
$\tilde f|_{U_i}$
, and hence of
$f|_{P_i}$
, so that
$\|D\phi _i\|\leq \unicode{x3bb} <1$
for all
$i\in \{1,2,\ldots ,m\}$
. We denote a
${\mathcal C}^3$
IFS associated to f by
$$ \begin{align*} \Phi_{\! f}=\{\phi_1,\phi_2,\ldots,\phi_m\}. \end{align*} $$
Remark 4.5. Unfortunately, it is not obvious if one can obtain an extension in which
$\phi _i$
are invertible. This would simplify the following arguments as one would not have to struggle to restrict the discussion to the sets
$U_i$
(e.g., see (4.11)). However, since
${\mathcal C}^\infty ({\mathbb R}^d,{\mathbb R}^d)$
finite to one maps are generic by Tougeron’s theorem (see [[Reference Golubitsky and Guillemin15], Theorem 2.6, pp. 169]), we can assume, by an arbitrarily small perturbation of f, that the
$\phi _i$
are finite to one.
For
$m,n\in {\mathbb N}$
, let
$\Sigma ^m_n=\{1,2,\ldots ,m\}^n$
and
$\Sigma ^m=\{1,2,\ldots ,m\}^{{\mathbb N}}$
be the standard symbolic spaces. We endow
$\Sigma ^m$
with the metric
$d_\gamma $
for some
$\gamma>1$
:
$$ \begin{align} d_{\gamma}(\sigma, \sigma^{\prime})=\sum_{i=1}^{\infty}\frac{|\sigma_i-\sigma^{\prime}_i|}{\gamma^i}. \end{align} $$
In addition, let
$\tau :\Sigma ^m\to \Sigma ^m$
be the left subshift:
$\tau (\sigma _1,\sigma _2,\sigma _3, \ldots )= (\sigma _2,\sigma _3,\ldots )$
.
Set
$K\kern1.3pt{=}\kern1.3pt\max \{\|x\|\;:\;x\in X\}$
(
$\|\cdot \|$
is the general Euclidean norm). Let
$M\kern1.3pt{=}\kern1.3pt\sup _i\|\phi _i(0)\|$
. Then, for each
$y\in {\mathbb R}^d$
,
$$ \begin{align*} \|\phi_i(y)\|\leq \|\phi_i(y)-\phi_i(0)\|+M\leq \unicode{x3bb} \|y\|+M. \end{align*} $$
Thus, setting
$$ \begin{align} Y=\{y\in {\mathbb R}^d\;:\; \|y\|\leq \max\{K,(1-\unicode{x3bb})^{-1} M\}\}, \end{align} $$
we have
$\phi _i(Y)\subset Y$
, for all
$i\in \{1,\ldots , m\}$
,
$X\subset Y $
and Y is a d-dimensional manifold with boundary.
Next, define
$\Theta _{\! f}:\Sigma ^m\to Y$
as
$$ \begin{align} \Theta_{\! f}(\sigma)=\bigcap\limits_{n\in{\mathbb N}} {\phi_{\sigma_1}\circ \phi_{\sigma_2}\circ \cdots \circ \phi_{\sigma_n}(Y)}, \end{align} $$
where
$\sigma =(\sigma _1, \sigma _2,\ldots ,)\in \Sigma ^m$
. The sets
$\{\phi _{\sigma _1}\circ \phi _{\sigma _2}\circ \cdots \phi _{\sigma _n}(Y)\}_{n\in {\mathbb N}}$
form a nested sequence of compact subsets of Y. In addition,
$\operatorname {diam}({\phi _{\sigma _1}\circ \phi _{\sigma _2}\circ \cdots \circ \phi _{\sigma _n}(Y)})\to 0$
as
$n\to \infty $
, so, by Cantor’s intersection theorem,
$\Theta _{\! f}(\sigma )$
is a single element in Y for every
$\sigma \in \Sigma $
, which implies that
$\Theta _{\! f}$
is well defined. We define the attractor of the IFS
$\Phi _{\! f}$
as
$$ \begin{align} \Lambda(\Phi_{\! f})=\Theta_{\! f}(\Sigma^m). \end{align} $$
Lemma 4.6. The function
$\Theta _{\! f}:\Sigma ^m\to Y$
is continuous. In turn,
$\Lambda (\Phi _{\! f})$
is compact.
Proof. For given
$\varepsilon>0$
, there exists
$k\in {\mathbb N}$
such that
$\unicode{x3bb} ^k\operatorname {diam}(Y)<\varepsilon $
. Let
$\delta =\gamma ^{-k}$
, where
$\gamma $
is the one in (4.1). For
$\sigma ,\sigma ^\prime \in \Sigma ^m$
,
$d_{\gamma }(\sigma , \sigma ^{\prime })<\delta =\gamma ^{-k}$
implies that, for all
$i<k$
,
$\sigma _i\!=\sigma ^{\prime }_i$
, and hence, for any
$x,y\in Y$
,
$$ \begin{align*} & d_0(\Theta_{\! f}(\sigma), \Theta_{\! f}(\sigma^\prime)) \\ &\quad \leq\sup_{x,y\in Y} d_0(\phi_{\sigma_1}\circ\cdots\circ\phi_{\sigma_k}\circ\phi_{\sigma_{k+1}}\cdots\circ\phi_{\sigma_n}(x), \\ &\qquad\qquad\qquad \phi_{\sigma_1}\circ\cdots\circ\phi_{\sigma_k}\circ\phi_{\sigma^{\prime}_{k+1}}\cdots\circ \phi_{\sigma^{\prime}_n}(y))\\ &\quad <\unicode{x3bb}^k \sup_{x,y\in Y} d_0(\phi_{\sigma_{k+1}}\circ\cdots\circ\phi_{\sigma_n}(x),\phi_{\sigma^{\prime}_{k+1}}\cdots\circ \phi_{\sigma^{\prime}_n}(y))\\ &\quad <\unicode{x3bb}^k \operatorname{diam}(Y)<\varepsilon, \end{align*} $$
and hence the continuity with respect to
$\sigma $
. Since
$\Sigma ^m$
is compact, the attractor
$\Lambda (\Phi _{\! f})$
is compact as it is the continuous image of a compact set.
Remark 4.7. Let
$\Phi _{\! f}=\{\phi _1,\ldots ,\phi _m\}$
be an IFS associated to a piecewise contraction f. For
$p\in {\mathbb N}$
, let
${\boldsymbol P}(f^p)=\{P^p_1,P^p_2,\ldots , P^p_{m_p}\}$
be the partition of
$f^p$
given as in equation (2.1). Define the corresponding IFS associated to
$f^p$
as
$$ \begin{align*} \Phi_{f^p}=\{\varphi_1,\varphi_2,\ldots,\varphi_{m_p}\}, \end{align*} $$
where, for all
$i\in \{1,2,\ldots ,m_p\}$
, there exists unique
$\sigma ^i=(\sigma ^i_1,\ldots ,\sigma ^i_p)\in \Sigma _p^m$
(uniquely determined by the partition element
$P_i^p\in {\boldsymbol P}(f^p)$
) such that
$$ \begin{align*} \varphi_i=\phi_{\sigma_1^i}\circ \phi_{\sigma_2^i}\circ\cdots\circ\phi_{\sigma_p^i}. \end{align*} $$
The attractor of
$\Phi _{f^p}$
is
$\Lambda (\Phi _{f^p})= \Theta _{\! f^p}(\Sigma ^{m_p})$
. To avoid confusion, we denote the elements in
$\Sigma ^m$
by
$\sigma $
and the elements in
$\Sigma ^{m_p}$
by
$\omega $
.
Lemma 4.8. For a piecewise smooth contraction f with IFS
$\Phi _{\! f}$
and for
$p\in {\mathbb N}$
, the following relationship holds.
$$ \begin{align*} \Lambda(f)\subset\Lambda(f^p)\subset \Lambda(\Phi_{f^p})\subset \Lambda(\Phi_{\! f}). \end{align*} $$
Proof. For
$p\in {\mathbb N}$
, let
${\boldsymbol P}(f^p)=\{P_1^p,P_2^p,\ldots ,P_{m_p}^p\}$
and let
$m_1=m$
. We start by proving the first inclusion, that is
$\Lambda (f)\subset \Lambda (f^p)$
. Let
$x\in \Lambda (f)= \bigcap _{n\in {\mathbb N}}\overline {{f}^n(X)}$
, that is, for every
$n\in {\mathbb N}$
,
$x\in \overline {f^n(X)}$
. Accordingly,
$$ \begin{align*} x\in \bigcap_{n\in{\mathbb N}}\overline{(f^p)^n(X)}= \Lambda(f^p). \end{align*} $$
Thus,
$\Lambda (f)\subset \Lambda (f^p)$
. For the third inclusion, that is,
$\Lambda (\Phi _{f^p})\subset \Lambda (\Phi _{\! f})$
, let
$\Phi _{f^p}=\{\varphi _1,\varphi _2,\ldots ,\varphi _{m_p}\}$
. Then, for every
$\omega =(\omega _1,\omega _2,\ldots )\in \Sigma ^{m_p}$
there exists
$\sigma ^{\omega }=(\sigma ^{\omega _1},\sigma ^{\omega _2}, \ldots )$
, where
$\sigma ^{\omega _i}=(\sigma ^{\omega _i}_1,\ldots ,\sigma ^{\omega _i}_p)\in \Sigma ^{m}_p$
, such that
$$ \begin{align*} \Theta_{\! f^p}(\omega)=\bigcap_{n\in{\mathbb N}}\varphi_{\omega_1}\circ\varphi_{\omega_2}\circ\cdots\circ\varphi_{\omega_n}(Y)=\bigcap_{n\in{\mathbb N}} \phi_{\sigma^{\omega_1}_1}\circ\cdots\circ\phi_{\sigma^{\omega_1}_p}\circ\cdots\circ\phi_{\sigma^{\omega_n}_p}(Y). \end{align*} $$
Thus,
$\Lambda (\Phi _{f^p})\!=\! \bigcup _{\omega \in \Sigma ^{m_p}}\Theta _{\! f^p}(\omega )\!=\!\bigcup _{\{\sigma ^\omega \;:\;\omega \in \Sigma ^{m_p}\}}\Theta _{\! f}(\sigma ^\omega )\subset \bigcup _{\sigma \in \Sigma ^{m}}\Theta _{\! f}(\sigma )=\Lambda (\Phi _{\! f}) $
. Finally, for the second inclusion, let
$x\in \Lambda (f^p)=\bigcap _{n\in {\mathbb N}}\overline {f^{pn}(X)}$
. Then, for all
$n\in {\mathbb N}$
, there exists
$y_n\in X$
such that
$d_0(x,f^{pn}(y_n))<1/n$
. By definition of
$\Phi _{f^p}$
, there exists
$\omega ^n=(\omega _1^n,\omega _2^n,\ldots ,\omega _n^n,\ldots )\in \Sigma ^{m_p}$
such that
$f^{pn}(y_n)=\varphi _{\omega _1^n}\circ \cdots \circ \varphi _{\omega _n^n}(y_n)$
, where
$\varphi _{\omega _i^n}\in \Phi _{f^p}$
. This implies that, for all
$n\in {\mathbb N}$
,
$d_0(x,\varphi _{\omega _1^n}\circ \cdots \circ \varphi _{\omega _n^n}(y_n))<1/n$
. By compactness of
$\Sigma ^{m_p}$
, there exists a subsequence
$\{n_k\}$
and
$\omega \in \Sigma ^{m_p}$
such that
$\omega ^{n_k}\to \omega $
. Since, by Lemma 4.6,
$\Theta _{\! f^p}$
is continuous,
$$ \begin{align*} d_0(x, \Theta_{\! f^p}(\omega))= \lim_{k\to\infty} d_0(x, \Theta_{\! f^p}(\omega^{n_k}))=\lim_{k\to\infty} d_0(x,f^{pn_k}(y_{n_k}))=0. \end{align*} $$
Hence, by definition of the attractor,
$x\in \Lambda (\Phi _{f^p})$
.
4.2. A simple perturbation and the proof of Proposition 4.2
For
$\delta \in {\mathbb R}^d$
with
${|\delta |>0}$
sufficiently small, and a piecewise contraction f with IFS
$\Phi _{\! f}=\{\phi _1,\phi _2,\ldots ,\phi _m\}$
, we define perturbations
$f^{\delta }, \Phi _{f^\delta }$
as
$$ \begin{align} f^{\delta}(x)=f(x)+\delta,\ \phi_i^{\delta}=\phi_i+\delta. \end{align} $$
Provided
$|\delta |$
is small enough, the perturbation
$f^\delta $
satisfies
$\overline {f^\delta (X)}\subset \mathring X$
, and hence
$f^\delta $
is a piecewise smooth contraction with corresponding IFS
$\Phi _{f^\delta }=\{\phi ^\delta _1,\phi ^\delta _2,\ldots ,\phi ^\delta _m\}$
. One can easily check that
$d_2(f,f^\delta )=|\delta |$
. By definition (see (4.3)),
$\Theta _{\! f^\delta }:\Sigma ^m\to Y$
reads
$$ \begin{align*} \Theta_{\! f^{\delta}}(\sigma)= \bigcap_{n \in{\mathbb N}}{\phi_{\sigma_1}^{\delta}\circ\phi_{\sigma_2}^{\delta}\circ\cdots\circ\phi_{\sigma_n}^{\delta}(Y)} \end{align*} $$
and the respective attractor is
${\Lambda }(\Phi _{f^\delta })= \bigcup _{\sigma \in \Sigma ^m}\Theta _{\! f^{\delta }}(\sigma )$
.
Observe that, for any
$p\in {\mathbb N}$
, the corresponding IFS associated to
$(f^{\delta })^p$
is given by
$\Phi _{(f^\delta )^p}=\{\varphi _1^\delta , \varphi _2^\delta , \ldots , \varphi _{m_p}^\delta \}$
, where, for every
$i\in \{1,2,\ldots ,m_p\}$
, there exists
$\sigma _i\in \Sigma ^m_p$
such that
$\varphi _i^\delta = \phi _{\sigma _1^i}^\delta \circ \phi _{\sigma _2^i}^\delta \circ \cdots \circ \phi _{\sigma _p^i}^\delta $
.
Lemma 4.9. The map
$\Theta _{\! f^{\delta }}(\sigma )\mapsto \bigcap _{n \in {\mathbb N}}{\phi _{\sigma _1}^{\delta }\circ \phi _{\sigma _2}^{\delta }\circ \cdots \circ \phi _{\sigma _n}^{\delta }(Y)}$
is uniformly Lipschitz continuous in
$\delta $
: that is, there exists
$a>0$
such that, for all
$\sigma \in \Sigma ^m$
,
$d_0(\Theta _{\! f^{\delta }}(\sigma ), \Theta _{\! f^{\delta ^{\prime }}}(\sigma )) \leq a d_0(\delta ,\delta ^{\prime })$
.
Proof. Let
$\delta ,\delta ^\prime>0$
,
$n\in {\mathbb N}$
,
$x\in Y$
and
$\sigma =(\sigma _1,\sigma _2,\ldots ,\sigma _n,\ldots )$
. Then
$$ \begin{align*} d_0(\Theta_{\! f^{\delta}}(\sigma), \Theta_{\! f^{\delta^\prime}}(\sigma))&\leq d_0(\Theta_{\! f^{\delta}}(\sigma),\phi_{\sigma_1}^{\delta}\circ \cdots \circ \phi_{\sigma_n}^{\delta}(x) )\\ &\quad +d_0(\Theta_{\! f^{\delta^\prime}}(\sigma),\phi_{\sigma_1}^{\delta^{\prime}}\circ \cdots \circ \phi_{\sigma_n}^{\delta^{\prime}}(x))\\ &\quad +d_0(\phi_{\sigma_1}^{\delta}\circ \cdots \circ \phi_{\sigma_n}^{\delta}(x),\phi_{\sigma_1}^{\delta^{\prime}}\circ \cdots \circ \phi_{\sigma_n}^{\delta^{\prime}}(x))\\ & \leq \unicode{x3bb}^nd_0(\Theta_{\! f^{\delta}}(\tau^n\sigma),x)+ \unicode{x3bb}^n d_0(\Theta_{\! f^{\delta^\prime}}(\tau^n\sigma),x)\\ &\quad +d_0(\delta,\delta^{\prime}){\kern-1pt} +{\kern-1pt}d_0(\phi_{\sigma_1}{\kern-1pt}\circ{\kern-1pt} \phi_{\sigma_2}^{\delta}{\kern-1pt}\circ \cdots \circ \phi_{\sigma_n}^{\delta}(x),\phi_{\sigma_1}{\kern-1pt}\circ \phi_{\sigma_2}^{\delta^\prime}{\kern-1pt}\circ \cdots \circ \phi_{\sigma_n}^{\delta^{\prime}}(x))\\ & \leq 2\unicode{x3bb}^n\operatorname{diam}(Y) +d_0(\delta,\delta^{\prime})\\&\quad +\unicode{x3bb} d_0(\phi_{\sigma_2}^{\delta}\circ \cdots \circ \phi_{\sigma_n}^{\delta}(x),\phi_{\sigma_2}^{\delta^\prime}\circ \cdots \circ \phi_{\sigma_n}^{\delta^{\prime}}(x)). \end{align*} $$
Iterating the above argument yields
$$ \begin{align*} \begin{split} d_0(\Theta_{\! f^{\delta}}(\sigma), \Theta_{\! f^{\delta^\prime}}(\sigma)) &\leq \lim_{n\to\infty}\{2\unicode{x3bb}^n\operatorname{diam}(Y)+ d_0(\delta,\delta^{\prime})(1+\unicode{x3bb}+\unicode{x3bb}^2+\cdots+ \unicode{x3bb}^n)\}\\ &=(1-\unicode{x3bb})^{-1}d_0(\delta,\delta^{\prime}), \end{split} \end{align*} $$
and letting
$a=1/(1-\unicode{x3bb} )$
concludes the proof.
Proof of Proposition 4.2
Let
$p\in {\mathbb N}$
be such that
$m_p\unicode{x3bb} ^p\leq \tfrac 12$
. Lemma 4.8 asserts that
$\Lambda (f^\delta )\subset \Lambda (\Phi _{(f^\delta )^p})$
. Hence, by Theorem 2.8, it suffices to prove that, for every
$\varepsilon>0$
small enough, there exists
$\delta \in B_{\varepsilon }(0)$
such that the attractor
$\Lambda (\Phi _{(f^\delta )^p})$
is disjoint from
$\partial {\boldsymbol P}(f^\delta )=\partial {\boldsymbol P}(f)$
.
Suppose, to the contrary, that, for every
$\delta \in B_{\varepsilon }(0)$
,
$\Lambda (\Phi _{(f^\delta )^p})\cap \partial {\boldsymbol P}(f)\neq \emptyset $
. Accordingly, there exists
$\omega (\delta )\in \Sigma ^{m_p}$
for which
$\Theta _{\! (f^\delta )^{p}}(\omega (\delta ))\in \partial {\boldsymbol P}(f)$
. By definition,
$\partial {\boldsymbol P}(f)=\bigcup _{P\in {\boldsymbol P}(f)}\partial P$
. Therefore, there exists
$P_i\in {\boldsymbol P}(f)$
and
$A\subset B_{\varepsilon }(0)$
with
$\mu _d(A)\geq \mu _d(B_{\varepsilon }(0))/m = C_d\varepsilon ^d/m$
(
$\mu _d$
is the d-dimensional Lebesgue measure) such that, for all
$\delta \in A,\ \Theta _{\! (f^\delta )^p}(\omega (\delta ))\in \partial P_i $
.
Moreover, for each
$k\in {\mathbb N}$
, there exist
$\omega ^*= (\omega ^*_1, \omega ^*_2,\ldots ,\omega ^*_k)\in \Sigma ^{m_p}_k$
such that the set defined as
$$ \begin{align*} A_k(\omega^*)= \{\delta\in A: \omega(\delta)_j=\omega^*_j,j\leq k\} \end{align*} $$
is non-empty and
$\mu _d(A_k(\omega ^*))\geq \mu _d(A)/m_p^k\geq C_d\varepsilon ^dm^{-1}{m_p}^{-k}$
. Accordingly, for
$\omega (\delta )\in A_k(\omega ^*)$
and the IFS associated to
$\Phi _{(f^\delta )^p}=\{\varphi _1^\delta ,\varphi _2^\delta ,\ldots ,\varphi _{m_p}^\delta \}$
,
$$ \begin{align*} \partial P_i\ni \Theta_{\! (f^\delta)^p}(\omega(\delta))&=\varphi_{\omega^*_1}^{\delta}\circ\Theta_{\! (f^\delta)^p}(\tau \omega(\delta))= \varphi_{\omega^*_1}^{\delta}\circ\varphi_{\omega^*_2}^{\delta}\circ\cdots \circ\varphi_{\omega^*_k}^{\delta}\circ\Theta_{\! (f^\delta)^p}(\tau^k\omega(\delta)), \end{align*} $$
where
$\tau $
is the left shift as defined above and
$\varphi _{\omega _j^*}^\delta = \phi ^\delta _{\sigma ^p_{j,1}}\circ \phi _{\sigma ^p_{j,2}}^\delta \circ \cdots \circ \phi _{\sigma ^p_{j,p}}^\delta $
, for
$\omega ^*_j=(\sigma ^p_{j,1},\sigma ^p_{j,2},\ldots ,\sigma ^p_{j,p})\in \Sigma ^m_p$
with
$\phi ^\delta _{\sigma ^p_{j,s}}\in \Phi _{f^\delta }$
.
Next, for some
$\bar {x}\in X$
, define
$\theta :A_k(\omega ^*)\to X$
as
$\theta (\delta )=\varphi _{\omega ^*_1}^{\delta }\circ \varphi _{\omega ^*_2}^{\delta }\circ \cdots \circ \varphi _{\omega ^*_k}^{\delta }(\bar {x})$
. Then
$$ \begin{align*} d_0(\Theta_{\! (f^\delta)^p}(\omega(\delta)),\theta(\delta))\leq\unicode{x3bb}^{pk}d_0(\Theta_{\! (f^\delta)^p}(\tau^k\omega(\delta)),\bar{x})\leq \operatorname{diam}(Y)\unicode{x3bb}^{pk}=:B_*\unicode{x3bb}^{pk}. \end{align*} $$
Thus,
$\theta (\delta )$
belongs to a
$B_*\unicode{x3bb} ^{pk}$
neighbourhood of
$\partial P_i$
. Since
$\partial P_i$
is contained in the union of finitely many
${\mathcal C}^2$
manifolds, the Lebesgue measure of a
$B_*\unicode{x3bb} ^{pk}$
neighbourhood of
$\partial P_i$
is bounded above by
$C\unicode{x3bb} ^{pk} \mu _{d-1}(\partial P_i)$
for a fixed constant
$C>0$
. Accordingly,
$$ \begin{align} \mu_d(\theta(A_k(\omega^*)))\leq C\unicode{x3bb}^{pk} \mu_{d-1}(\partial P_i). \end{align} $$
On the other hand,
$$ \begin{align*} \mu_d(\theta(A_k(\omega^*)))=\int_{A_k(\omega^*)}|\kern-1pt \det(D\theta(\delta))|\,d\delta, \end{align*} $$
where, by definition of
$\theta (\delta )$
, 
. Note that
$$ \begin{align*} \|D\varphi_{\sigma_1^*}+D\varphi_{\sigma_1^*}D\varphi_{\sigma_2^*}+\cdots+D\varphi_{\sigma_1^*}\cdots D\varphi_{\sigma_{k-1}^*}\hspace{-1pt}\|\leq \frac{\unicode{x3bb}^p}{1-\unicode{x3bb}^p}, \end{align*} $$
where
$\|\cdot \|$
is the standard operator norm defined as
$\|L\|=\sup _{\|v\|=1}\|Lv\|$
for any linear operator
$L:{\mathbb R}^d\to {\mathbb R}^d$
. It follows that, for all
$v\in {\mathbb R}^d$
,
$$ \begin{align*} \|D\theta(\delta)v\|\geq \|v\|-\frac{\unicode{x3bb}^p}{1-\unicode{x3bb}^p}\|v\|\geq\frac{1}{2}\|v\|, \end{align*} $$
since
$\unicode{x3bb} ^p\leq {1}/{2m^p}$
and
$m^p\geq 2$
. Hence, the eigenvalues of
$D\theta (\delta )$
are larger, in modulus, than
$\tfrac 12$
. Accordingly,
$|\kern-2.5pt \det (D\theta (\delta ))|\geq 2^{-d}$
and
$$ \begin{align} \mu_d(\theta(A_k(\omega^*)))\geq 2^{-d} \mu_d(A_k(\omega^*))\geq C_d 2^{-d}\varepsilon^dm^{-1}{m_p}^{-k}, \end{align} $$
which, for k large enough, is in contradiction with (4.6), and this concludes the proof.
4.3. Fixed points in a generic position
Fix
$N\in {\mathbb N}$
. For all
$q\leq N+1$
and
${\sigma =(\sigma _1,\ldots ,\sigma _q)\in \Sigma ^m_{q}(\Phi )}$
, let
$x_\sigma (\Phi )$
be the unique fixed point of
$\phi _{\sigma _1}\circ \cdots \circ \phi _{\sigma _q}$
: that is,
$$ \begin{align} \phi_{\sigma_1}\circ\cdots\circ \phi_{\sigma_q}(x_\sigma(\Phi))=x_\sigma(\Phi). \end{align} $$
The goal of this section is to define a perturbation that puts the above fixed points in a generic position. We start with the following trivial but useful fact concerning the location of such fixed points.
Lemma 4.10. Given an IFS
$\Phi $
, if for some
$y\in {\mathbb R}^d, \delta>0$
,
$p\in {\mathbb N}$
and
$\sigma \in \Sigma ^m_p$
,
$$ \begin{align*} \phi_{\sigma_1}\circ\cdots\circ \phi_{\sigma_p}(B_\delta(y))\cap B_\delta(y)\neq \emptyset, \end{align*} $$
then the unique fixed point of
$\phi _{\sigma _1}\circ \cdots \circ \phi _{\sigma _p}$
belongs to
$ B_{c_*\delta }(y)$
,
$c_*= {2}/({1-\unicode{x3bb} })>2$
.
Proof. The fact that
$\|\phi _{\sigma _1}\circ \cdots \circ \phi _{\sigma _p}(y)-y\|\leq 2\delta $
implies that, for each
$x\in B_{c_*\delta }(y)$
,
$$ \begin{align*} \begin{split} \|\phi_{\sigma_1}\circ\cdots\circ \phi_{\sigma_p}(x)-y\|&\leq \|\phi_{\sigma_1}\circ\cdots\circ \phi_{\sigma_p}(x)-\phi_{\sigma_1}\circ\cdots\circ \phi_{\sigma_p}(y)\|+2\delta\\ &\leq (\unicode{x3bb}^p c_*+2)\delta\leq c_*\delta. \end{split} \end{align*} $$
Hence,
$\phi _{\sigma _1}\circ \cdots \circ \phi _{\sigma _p}(B_{c_*\delta }(y))\subset B_{c_*\delta }(y)$
. The lemma follows by the contraction mapping theorem.
Let
${\mathcal A}_p=\{\phi _{\sigma _{1}}\circ \cdots \circ \phi _{\sigma _p}\;:\;\sigma \in \Sigma ^{m}_{p}\}$
and
$A_p(\Phi ):=\{x_\sigma (\Phi )\;:\; \sigma \in \Sigma ^m_{p}(\Phi )\}$
and let
$\#A_{N}(\Phi )=k_\star $
. We can now explain what we mean by having the fixed points in a generic position.
Proposition 4.11. For each
$\varepsilon>0$
, there exists an
$\varepsilon $
-perturbation
$\Phi ^0=\{\phi ^0_k\}$
of
$\Phi $
such that, for all
$q\leq p\leq N$
,
$\sigma \in \Sigma ^m_{q}(\Phi )$
,
$\omega \in \Sigma ^m_{p}(\Phi )$
,
$\omega \neq \sigma $
, if
$\sigma _{p+1}\neq \omega _q$
, then
$x_{\sigma }(\Phi ^0)\neq x_{\omega }(\Phi ^0)$
, whereas if
$\sigma _{p+1}=\omega _q$
and
$x_{\sigma }(\Phi ^0)= x_{\omega }(\Phi ^0)$
, then
$\phi _{\sigma _{1}}\circ \cdots \circ \phi _{\sigma _q}$
and
$\phi _{\omega _{1}}\circ \cdots \circ \phi _{\omega _p}$
are both some power of a
$\Theta \in \bigcup _{s=1}^p {\mathcal A}_s$
.
Proof. We proceed by induction on p. If
$\sigma _1,\omega _1\in \{1,\ldots ,m\}$
and
$x_{\sigma _1}(\Phi )=x_{\omega _1}(\Phi )$
, then we can simply make the perturbation
$\tilde \phi _{\sigma _1}(x)=\phi _{\sigma _1}(x)+\eta $
for some
$\|\eta \|<\varepsilon /2$
. This proves the statement for
$p=1$
. To simplify the notation, we keep calling
$\Phi $
also the perturbed IFS. We suppose that the statement is true for p, after a perturbation of size at most
$(1-2^{-p})\varepsilon $
, and we prove it for
$p+1$
.
Since
$A(p+1)=\{x_\sigma (\Phi )\;:\; \sigma \in \Sigma ^m_{p+1,*}(\Phi )\}$
is a finite discrete set,
$$ \begin{align*} \delta^*_{p+1}=\min\Big\{1,\inf_{\substack{x,y\in A(p+1)\\ x\neq y}}\|x-y\|\Big\}>0. \end{align*} $$
Let
$\delta \leq c_*^{-2}\delta ^*_{p+1}/2$
(recall that
$c_*= {2}/({1-\unicode{x3bb} })$
). Suppose that
$z:=x_\sigma (\Phi )=x_\omega (\Phi )$
and
$\omega _{q}=\sigma _{p+1}$
, where
$\sigma \in \Sigma ^m_{p+1}(\Phi )$
,
$\omega \in \Sigma ^m_{q}(\Phi )$
,
$q\leq p+1$
and
$\sigma \neq \omega $
. Let
${j\in \{0,\ldots , q-1\}}$
be the largest integer such that
$\sigma _{p+1-j}=\omega _{q-j}$
. If
$j=q-1$
, then it must be that
$q\leq p$
; otherwise, we would have
$\omega =\sigma $
. Hence,
$$ \begin{align*} \phi_{\omega_{1}}\circ \cdots\circ\phi_{\omega_q}(z)\hspace{-0.5pt}=\hspace{-0.5pt}z\hspace{-0.5pt}=\hspace{-0.5pt}\phi_{\sigma_1}\circ \cdots \circ\phi_{\sigma_{p+1}}(z)\hspace{-0.5pt}=\hspace{-0.5pt}\phi_{\sigma_1}\circ \cdots \circ\phi_{\sigma_{p+1-q}}\circ \phi_{\omega_{1}}\circ \cdots\circ\phi_{\omega_q}(z). \end{align*} $$
That is,
$\phi _{\sigma _1}\circ \cdots \circ \phi _{\sigma _{p+1-q}}(z)=z$
. It follows, by the inductive hypothesis, that there exist
$k\in {\mathbb N}$
and
$\Theta \in {\mathcal A}_s$
,
$s\leq q$
, such that
$$ \begin{align*} \begin{split} & \phi_{\sigma_{1}}\circ \cdots \circ\phi_{\sigma_{p+1-q}}=\Theta^k,\\ &\phi_{\omega_{1}}\circ \cdots\circ\phi_{\omega_q}=\Theta^j, \end{split} \end{align*} $$
so
$\phi _{\sigma _1}\circ \cdots \circ \phi _{\sigma _{p+1}}=\Theta ^{k+j}$
, as claimed. It remains to consider the case
$j<q-1$
. Let
$\psi := \phi _{\omega _{q-j}}\circ \cdots \circ \phi _{\omega _q}$
. Then
$\psi =\phi _{\sigma _{p+1-j}}\circ \cdots \circ \phi _{\sigma _{p+1}}$
and
$$ \begin{align*} [\psi\circ \phi_{\omega_1}\circ \cdots \circ \phi_{\omega_{q-j-1}}]( \psi(z))=\psi(z)= [\psi\circ \phi_{\sigma_1}\circ \cdots \circ\phi_{\sigma_{p-j}}](\psi(z)), \end{align*} $$
where, by construction,
$\omega _{q-j-1}\neq \sigma _{p-j}$
. By renaming the indices, we are thus reduced to the case
$\omega _{q}\neq \sigma _{p+1}$
. The following lemma is useful for analysing this case.
Sub-lemma 4.12. If, for
$j\in \{1,\ldots , p\}$
,
$$ \begin{align*} \phi_{\sigma_{j+1}}\circ\cdots \circ \phi_{\sigma_{p+1}}(B_\delta(z))\cap B_\delta(z)\neq \emptyset, \end{align*} $$
then
$\phi _{\sigma _{j+1}}\circ \cdots \circ \phi _{\sigma _{p+1}}(z)=z$
and
$\sigma _j=\sigma _{p+1}$
, and the same for
$\omega $
.
Proof. Lemma 4.10 implies that there exists
$z_1\in B_{c_*\delta }(z)$
such that
$\phi _{\sigma _{j+1}}\circ \cdots \circ \phi _{\sigma _{p+1}}(z_1)=z_1$
. In addition,
$$ \begin{align*} \|z-\phi_{\sigma_{1}}\circ\cdots \circ \phi_{\sigma_{j}}(z_1)\|=\|\phi_{\sigma_{1}}\circ\cdots \circ \phi_{\sigma_{p+1}}(z)-\phi_{\sigma_{1}}\circ\cdots \circ \phi_{\sigma_{p+1}}(z_1)\|\leq\unicode{x3bb} c_*\delta. \end{align*} $$
Thus,
$\phi _{\sigma _{1}}\circ \cdots \circ \phi _{\sigma _{j}}(B_{c_*\delta }(z))\cap B_{c_*\delta }(z)\neq \emptyset $
, and hence Lemma 4.10 implies that there exists
$z_2\in B_{c_*^2\delta }(z)$
such that
$\phi _{\sigma _{1}}\circ \cdots \circ \phi _{\sigma _{j}}(z_2)=z_2$
. The definition of
$\delta $
implies that
$z_1=z_2$
, which, in turn, implies that
$z_1=z$
. Hence,
$$ \begin{align*} &\phi_{\sigma_{1}}\circ\cdots \circ \phi_{\sigma_{j}}(z)=z,\\ &\phi_{\sigma_{j+1}}\circ\cdots \circ \phi_{\sigma_{p+1}}(z)=z, \end{align*} $$
and, by the inductive hypothesis, this is possible only if
$\sigma _{j}=\sigma _{p+1}$
. The argument for
$\omega $
is identical.
If there exists
$j\in \{1,\ldots , p\}$
and
$k\in \{1,\ldots , q\}$
such that
$$ \begin{equation}\begin{aligned} &\phi_{\sigma_{j+1}}\circ\cdots \circ \phi_{\sigma_{p+1}}(B_\delta(z))\cap B_\delta(z)\neq \emptyset,\\ &\phi_{\omega_{k+1}}\circ\cdots \circ \phi_{\omega_{q}}(B_\delta(z))\cap B_\delta(z)\neq \emptyset, \end{aligned}\end{equation} $$
then, by Sub-Lemma 4.12,
$\phi _{\sigma _{j+1}}\circ \cdots \circ \phi _{\sigma _{p+1}}(z)=z=\phi _{\omega _{k+1}}\circ \cdots \circ \phi _{\omega _{q}}(z)$
, which contradicts our inductive hypothesis. Thus, if the first inequality is satisfied for some j, the second cannot be satisfied for any k, and vice versa. Suppose that there does not exist k for which the second inequality of (4.9) is satisfied (the other possibility being completely analogous).
Define the perturbation
$$ \begin{align*} \tilde\phi_k=\begin{cases} \phi_k\circ h_{z,\delta} &\mathrm{if}\ k=\omega_{q},\\ \phi_k&\mathrm{ otherwise,} \end{cases} \end{align*} $$
where, by analogy with (4.18), for
$v\in {\mathbb R}^d$
,
$\|v\|=1$
,
$$ \begin{align*} h_{z,\delta}(x)=\begin{cases} x\quad&\text{for all } x\not\in B_{c_*^{-1}\delta}(z),\\ x+c_*^{-3}\delta^3g(1-c_*\delta^{-1}\|x-z\|) v &\mathrm{otherwise}. \end{cases} \end{align*} $$
Note that
$\|h_{z,\delta }-\mathrm {id}\|_{{\mathcal C}^2}\leq 2^{-p-1}\varepsilon _0$
. Since
$h_{z,\delta }(B_{c_*^{-1}\delta }(z))\subset B_{c_*^{-1}\delta }(z)$
, it follows that the effect of the perturbation is always confined to
$B_{\delta }(z)$
and its images. Moreover, by Sub-Lemma 4.12, if there exists j such that
$\phi _{\sigma _{j+1}}\circ \cdots \circ \phi _{\sigma _{p+1}}(B_\delta (z))\cap B_\delta (z)\neq \emptyset $
, we have seen that
$\sigma _{j}=\sigma _{p+1}\neq \omega _q$
, so next we apply a map,
$\phi _{\sigma _j}$
, that has not been modified. On the other hand, if
$x\in B_\delta (z)$
and for some k we have
$\omega _k=\omega _{q}$
, we have
$\phi _{\omega _{k+1}}\circ \cdots \circ \phi _{\omega _q}(x)\not \in B_\delta (z)$
. Next, we apply
$\phi _{\omega _k}$
outside the region where it has been modified. It follows that, for each
$x\in B_\delta (z)$
,
$$ \begin{align*} \begin{split} & \tilde\phi_{\omega_{1}}\circ\cdots \circ \tilde\phi_{\omega_{q}}(x)=\phi_{\omega_{1}}\circ\cdots \circ \phi_{\omega_{q}}\circ h_{z,\delta}(x),\\ &\tilde \phi_{\sigma_{1}}\circ\cdots \circ \tilde \phi_{\sigma_{p+1}}(x)=\phi_{\sigma_{1}}\circ\cdots \circ \phi_{\sigma_{p+1}}(x). \end{split} \end{align*} $$
Calling
$z(\delta )$
the unique fixed point of
$\phi _{\omega _{1}}\circ \cdots \circ \phi _{\omega _{q}}\circ h_{z,\delta }$
,
Since
$z(0)=z$
and
$$ \begin{align*} \partial_\delta h_{z,\delta}=3c_*^{-3}\delta^2g(1-c_*\delta^{-1}\|z(\delta)-z\|) v +c_*^{-2}\delta \|z(\delta)-z\|g'(1-c_*\delta^{-1}\|z(\delta)-z\|) v, \end{align*} $$
we have that it is not possible that
$z(\delta )=z$
for all
$\delta \leq c_*^{-2}\delta ^*_{p+1}/2$
. Thus, we can make a perturbation for which the two fixed points are different, and, for
$\delta $
small, they cannot be equal to the other fixed points.
For any other couple of elements
$\sigma \in \Sigma ^m_{p+1}(\Phi ),\omega \in \Sigma ^m_{q}(\Phi )$
, we can repeat the same process and obtain the perturbation with two different fixed points, as above. Note that, as the size of the perturbation is
$\delta <c_*^{-1}\delta ^*_{p+1}/2$
, the distance between the newly obtained fixed points in
$\bigcup _{q\leq p}{\mathcal A}_q$
stays positive as the perturbation does not move the fixed points more than
$\delta ^*_{p+1}/2$
.
From now on, we assume that
$\Phi $
satisfies Proposition 4.11.
4.4. Pre-images of the boundary manifolds and how to avoid them
Next, we need some notation and a few lemmata to describe the structure of the pre-images of the discontinuity manifolds conveniently. This allows us to develop the tools to prove Proposition 4.3.
Let f be a piecewise smooth contraction with
${\boldsymbol P}(f)=\{P_1, P_2,\ldots , P_m\}$
and
${\Phi _{\! f}=\{\phi _1,\phi _2,\ldots ,\phi _m\}}$
. Recall that, by hypothesis,
$\partial {\boldsymbol P}(f)$
is contained in the finite union of
${\mathcal C}^2$
manifolds, which we will call boundary manifolds. Let
$l_0$
be the number of boundary manifolds in
$\partial {\boldsymbol P}(f)$
. Recall also that, for every
$i\in \{1,2,\ldots ,m\}$
,
$U_{i}$
is the open neighbourhood of
$P_{i}\in {\boldsymbol P}(f)$
such that
$\tilde f|_{U_{i}}$
is injective and hence invertible. Accordingly, by the construction of
$\Phi _{\! f}$
, we have that
$\phi _{i}|_{U_{i}}$
has a well-defined inverse for all
$i\in \{1,2,\ldots ,m\}$
.
Let
$\epsilon _0=\min \{ d_H(P_i,U_i^c)\;:\; i\in \{1,\ldots ,m\}\}$
, where the complement is taken in
${\mathbb R}^d$
. For each
$\epsilon \leq \epsilon _0/2$
, we can consider the
$\epsilon $
-neighbourhood
$V_i$
of
$P_i$
and the
$\epsilon /2$
neighbourhood
$V_i^-$
.
Choosing
$\epsilon $
small enough, we can describe the boundary manifolds by embeddings
$\psi _i\in {\mathcal C}^2(D_i^+, {\mathbb R}^d)$
,
$i\in \{1,2,\ldots ,l_0\}$
, such that
$\overline {\psi _i(D^+_i)}\subset U_{p}$
for some
$p\in \{1,\ldots ,m\}$
, and there exists an open set
$D_i\subset D^+_i\subset {\mathbb R}^{d-1} $
such that
$\psi _i(D_i)\cap V_{p}\neq \emptyset $
and
$\partial \psi _i(D_i)\cap \overline V_{p}=\emptyset $
(this is possible by Definition 2.16). For each IFS
$\Phi $
and
$\sigma \in \Sigma ^m_n$
, recalling the definition (4.2) of Y, we let
$$ \begin{equation}\begin{aligned} &D_{\sigma}(\Phi)=\{x\in Y\;:\; x\in V^-_{\sigma_n}, \phi_{\sigma_{k+1}}\circ\cdots\circ \phi_{\sigma_n}(x)\in V^-_{\sigma_{k}}, k\in \{1,\ldots, n-1\}\}, \\ &D^+_{\sigma}(\Phi)=\{x\in Y: x\in V_{\sigma_n}, \phi_{\sigma_{k+1}}\circ\cdots\circ \phi_{\sigma_n}(x)\in V_{\sigma_{k}}, k\in \{1,\ldots, n-1\}\}. \end{aligned}\end{equation} $$
We call a sequence
$\sigma $
admissible if
$D_{\sigma }(\Phi )\neq \emptyset $
. We define the set