On the Space of Iterated Function Systems and Their Topological Stability
aa r X i v : . [ m a t h . D S ] S e p ON THE SPACE OF ITERATED FUNCTION SYSTEMS ANDTHEIR TOPOLOGICAL STABILITY
A. ARBIETO, A. TRILLES
Abstract.
We study iterated function systems (IFS) with compact parameterspace. We show that the space of IFS with phase space X is the hyperspaceof the space of self continuous maps of X . With this result we obtain that theHausdorff distance is a natural metric for this space which we use to definetopological stability.Then we prove, in the context of IFS, the classical results showing thatshadowing property is a necessary condition for topological stability and shad-owing property added to expansiveness are a sufficient condition for topologicalstability. To prove these statements, in fact, we use a stronger type of shad-owing, called concordant shadowing property.We also give an example showing that concordant shadowing property istruly different than the traditional definition of shadowing property for IFS. Introduction
In 1981, Hutchinson [8] introduced the Iterated Function Systems (IFS) as away of studying fractals. In that case he studied only hyperbolic IFS with finiteparameter space, a finite collection of contractions. His theory and fractal theorywas disseminated by different books, as [4, 5].He realized that the study of the omega-limit set of a collection of maps isconnected with the iteration of compact sets. It is like a collective dynamics andhe found that the base space was already been study by topologists, the so calledhyperspace of a compact metric space, using the Hausdorff metric.Eventually, it was realized that the theory of IFS can be seen as the action ofa atomic measure on the space of dynamics over the phase space, generating arandom dynamical system. So, a natural question arises: Instead of an atomicmeasure (related with finitely maps) could be used a Radon measure with compactsupport? In other words, could be the parameter space compact instead of finite?This was pursuit by many authors as in [9, 11], and eventually Arbieto, Junqueirae Santiago [2] obtained several results in this setting assuming very weak sourcesof contractions. More recently, Melo [10] generalized this in his thesis.The theory of dynamical systems had a boost in mathematics with the adventof hyperbolic theory due to Smale [14]. The horseshoe became the paradigmaticexample and had two important topological dynamical features: expansiveness andthe shadowing. These two notions were extensively studied by many mathemati-cians such as Das, Kato, Sakai, Thakkar and many others. One of theirs best
Mathematics Subject Classification.
Primary: 37D20; Secondary: 37C70.
Key words and phrases.
Surface diffeomorphism, Homoclinic class, Axiom A.Partially supported by CNPq, FAPERJ and PRONEX/DS from Brazil. feature was that they were heavily used in the stability of hyperbolic differentiabledynamics, see [16].It turns out, that in topological dynamics, this also leads to some type of stability(nowadays called topological stability), see [1]. Moreover, it was shown that theshadowing property is a necessary condition to topological stability. This was aseminal result that gives rise to the study of shadowing-type properties and evenstronger forms of stability, like the Gromov-Hausdorff Stability by Arbieto andMorales, see [3].Naturally, this notion was exploited for IFS by some authors, see [13]. However,as far as we understand it is not quite precise. Moreover, the study is done in thefinite case. So, the purpose of this work is to clarify this issues and to prove in thecase of a compact parameter space.For this, we use a stronger type of shadowing for IFS and we propose anotherdefinition for topological stability. These definitions permit us to obtain our mainresults showing that this stronger shadowing is a necessary condition for topolog-ical stability, extending the original result for maps, which can be seen in [12], tothe context of IFS and we also show that this type of shadowing added to theexpansiveness are sufficient condition for topological stability.The notion of topological stability deals with proximity of objects, in our caseIFS. Looking for a way to measure distance between two IFS we notice that thespace of IFS with a fixed phase space X is the hyperspace of the space of selfcontinuous maps of X . With this result we obtain that the Hausdorff metric is anatural complete metric for the space of IFS.2. Definitions
Consider (
X, d ) a compact metric space and the space C ( X ) of the self contin-uous maps of X with the C -metric given by: d C ( f, g ) = max { d ( f ( x ) , g ( x ) : ∀ x ∈ X } A family { ω λ : λ ∈ Λ } ⊂ C ( X ) such that Λ is a compact metric space and ω : Λ × X → M , given by ω ( λ, x ) := ω λ ( x ) is a continuous map, is said to be an Iterated Function System (IFS for short), and we call ω its general map. The spaceΛ is called the parameter space and X is called the phase space of this IFS. We willoften refer to an IFS by its general map but it is important observe that differentgeneral maps can represent the same IFS.The space Λ N endowed with the product topology will be denoted by Ω. Foreach σ = ( λ , λ , ... ) ∈ Ω and k ∈ N we will denote the map ω σ k := ω λ k ◦ ... ◦ ω λ and ω σ := id . A sequence { x n } ⊂ X is called a chain (it can also be found as branch of orbit in the literature) for the IFS ω if for each n there exists λ n ∈ Λsuch that x n = ω λ n ( x n − ).Since any finite set with the discrete metric is a compact metric space, we observethat any IFS with finitely many partial maps is automatically included in ourdefinition.For ( X, d ) a metric space not necessarily compact, we will denote K ( X ) as thecollection of all nonempty compact subsets of X and call it the Hyperspace of X . Weendow it with the Hausdorff metric as follows. Let d ( x, F ) = inf { d ( x, y ) : y ∈ F } . N THE SPACE OF ITERATED FUNCTION SYSTEMS AND THEIR TOPOLOGICAL STABILITY3
The Hausdorff metric is given by: d H ( A, B ) = max (cid:26) sup a ∈ A d ( a, B ) , sup b ∈ B d ( b, A ) (cid:27) If (
X, d ) is a complete (resp. compact) metric space, it can be proved (see [4])that ( K ( X ) , d H ) is also a complete (resp. compact) metric space. With these factsit was possible to show that every hyperbolic IFS has a invariant attractor, in [2]the result was extended to weakly hyperbolic IFS and then after in [10] extendedthe result for P -weakly hyperbolic IFS.In our study we noticed that the IFS and the Hyperspace are even more related.Our first result in this work is to explicit the relation between them which is exposedin the next theorem. Theorem 1.
The space of IFS with phase space X is the Hyperspace of C ( X ) . This theorem permit us to conclude that the space of IFS with phase space X is a complete metric space with the Hausdorff metric, and it comes from the factthat since X is a compact metric space, then C ( X ) is a complete space with C -topology. To fix the notation, from now on, given ω : Λ × X → X and ˜ ω : ˜Λ × X → X both IFS, we will denote d H ( ω, ˜ ω ) := d H ( { ω λ : λ ∈ Λ } , { ˜ ω λ : λ ∈ ˜Λ } ).This topology on the space of IFS leads us to questions about density, opennessor even genericity of dynamical properties for IFS. As an example we can see thatthe set of transitive IFS is a G δ set.First of all, we say that an IFS is transitive if for any U, V open sets there is σ ∈ Ω and n such that ω σ n ( U ) ∩ V = ∅ . Let ω be a transitive IFS. If we fix U and V , for any ˜ ω in a sufficiently small neighborhood of ω , there is ˜ σ ∈ ˜Ω such that˜ ω ˜ σ n ( U ) ∩ V = ∅ . So using a countable basis of open sets of X we can conclude thatthe set of transitive IFS is a G δ set.3. Shadowing Property
For maps the shadowing property consists in guaranteeing the existence of anorbit close to a pseudo-orbit, which is a sequence similar to an orbit, but wheresmall errors is permitted for each iterate. This notion can be translated to thecontext of IFS trading orbit by chain.
Definition 3.1.
Given a sequence { x k } in X and δ > , this sequence is said tobe a δ -chain if for each k there exists λ k ∈ Λ such that d ( x k , ω λ k ( x k − )) ≤ δ . If thesequence is finite, we call it finite δ -chain. So, shadowing property in the context of IFS means that any ε > δ > δ -chain is ε -close to a chain. Definition 3.2.
We say that an IFS ω has the shadowing property if for any given ε > , there exists δ > such that for any δ -chain { x k } there exists a chain { y k } such that d ( x k , y k ) < ε for all k ≥ . In this case, we say that { y k } ( ε ) -shadows { x k } .Remark. We remark that this definitions of shadowing property does not guaran-tee any relation between the sequence of parameters of the shadow and the sequenceof parameters of the δ -chain. A. ARBIETO, A. TRILLES
In fact, we give an example which has shadowing property but for some δ -chainsit is impossible to shadow with the same sequence of parameters. Moreover, ourexample is constituted by rotations, so it is possible to have shadowing property inthe IFS even with all partial maps not having shadowing property.For some technical reasons, sometimes we would like to guarantee the existenceof a shadow with the same sequence of parameters of the δ -chain. For this we usea stronger definition of shadowing for IFS presented in [7]. Definition 3.3.
We say that an IFS has concordant shadowing property if for any ε > there exists δ > such that every δ -chain can be ε -shadowable by a chainwith the same sequence of parameters. In 2006, Glavan and Gutu [6] showed that this stronger definition is not sorestrictive despite not having used this term. They also worked in a more generalsetting, considering an IFS any collection of continuous maps.They start by the large and well known class of hyperbolic IFS (which can alsobe found as uniformly contracting IFS).
Definition 3.4.
We say that the an IFS { ω λ : λ ∈ Λ } is hyperbolic if sup λ ∈ Λ sup x = y d ( ω λ ( x ) , ω λ ( y )) d ( x, y ) < Theorem 3.5 (Glavan, Gutu [6]) . Every hyperbolic IFS has concordant shadowingproperty.
The second class studied by them goes in the opposite direction of the first one.In this case any pair of points move away uniformly from each other by one iterationof any partial map.
Definition 3.6.
We say that an IFS { ω λ : λ ∈ Λ } is uniformly expanding if inf λ ∈ Λ inf x = y d ( ω λ ( x ) , ω λ ( y ) d ( x, y ) > Theorem 3.7 (Glavan, Gutu [6]) . Every IFS uniformly expanding with all partialmaps being surjective has concordant shadowing property. Topological Stability
Topological Stability means essentially that the behavior of an IFS remains foranother IFS sufficiently close to the initial one, in a topological viewpoint.Differently than an orbit of a point for a map, for an IFS a point can haveinfinitely many chains, which together consist the orbit of the point. It is impos-sible compare any two chains between any different IFS, our goal is to analyzesimilar chains in similar IFS, and to be more precise we introduce the notion of δ -compatibility Definition 4.1.
For ω , ˜ ω two IFS and σ , ˜ σ sequences with the same length in eachparameter space, we say that the pair ( σ, ˜ σ ) is δ -compatible if for all k we have: d C ( ω λ k , ˜ ω ˜ λ k ) < δ Definition 4.2.
We say that an IFS ω is topologically stable if given ε > , thereexists δ > such that if ˜ ω is an IFS and d H ( ω, ˜ ω ) ≤ δ , then for each ( σ, ˜ σ ) δ -compatible there exists a continuous map h : X → X with the following properties: N THE SPACE OF ITERATED FUNCTION SYSTEMS AND THEIR TOPOLOGICAL STABILITY5 (i) d C ( ω σ k ◦ h, ˜ ω ˜ σ k ) < ε for all k ∈ N (ii) d C ( h, id ) < ε One of our main results, extending and clarifying the work of Rezaei and Niain [13] is to show that a consequence of topological stability is the shadowing prop-erty, actually we go further and we prove that topological stability implies con-cordant shadowing property for IFS having a smooth compact manifold as phasespace.
Theorem 2.
Every topologically stable IFS with a smooth compact manifold asphase space has concordant shadowing property.
For maps a converse for this theorem can be obtained by adding the hypothesisof expansiveness, we expected that this should be true for IFS too.Expansiveness, for maps, means essentially that for any two different points,their orbits move away from each other at least a constant. This notion can betranslated for IFS trading orbits for chains for any sequence of parameters.
Definition 4.3.
We say that an IFS ω is expansive if there exists a constant η > , called expansivity constant, such that for any σ ∈ Ω if x, y ∈ X satisfy d ( ω σ n ( x ) , ω σ n ( y )) ≤ η for all n ∈ N , then x = y . We usually say that ω is η − expansive . Similarly to what have been done for maps, we can prove that expansivenessadded to concordant shadowing property implies in the existence of a unique shadowfor a δ -chain where both of them have the same sequence of parameters. Using thisfact we construct for each pair of sequences δ -compatible a continuous map withthe properties desired for topological stability, proving the last result of this work. Theorem 3.
Every expansive IFS with concordant shadowing property is topolog-ically stable.Remark.
For the converse of the theorem the phase space is not required to bea manifold, it works for compact metric spaces in general.After give this proof, we observed that it is essentially the same of the proof givenby Thakkar and Das in [15] the observation is that δ -compatibility of sequences isequivalent to time varying maps δ -close for them.5. IFS of Homeomorphisms
Although during the text we are considering IFS for maps not necessarily in-vertible we can also consider
Homeo ( X ) the space of self homeomorphisms of X with the metric d h ( f, g ) = max (cid:8) d C ( f, g ) , d C ( f − , g − ) (cid:9) and replace C ( X ) by Homeo ( X ) in the theory.In this case we need to consider Ω = Λ Z ∗ , ω σ k ( x ) = ω − λ k ◦ ... ◦ ω − λ − for negative k and bilateral sequences instead of unilateral sequences in all definitions and in allour results.When we gave the definition of expansiveness, actually we presented the notion ofpositively expansive IFS. For IFS of homeomorphisms the definition is the following. Definition 5.1.
We say that an IFS ω is expansive if there exists a constant η > , called expansivity constant, such that for any σ ∈ Ω if x, y ∈ X satisfy d ( ω σ n ( x ) , ω σ n ( y )) ≤ η for all n ∈ Z then x = y . A. ARBIETO, A. TRILLES
Remark.
For shadowing property and topological stability if an IFS of homemo-rphisms satisfy the definitions for bilateral sequences, then it satisfies for unilateralsequences. For expansiveness it fails, since there are examples of IFS of homeomor-phisms which are expansive but they are not positively expansive.6.
Examples
During our study we asked ourselves if it was possible to guarantee the existenceof a shadow with the same sequence of parameters of the δ -chain in the traditionaldefinition of shadowing property. After some time we found the following exampleanswering negatively the question. Example 6.1.
Consider T = R / Z the unit circle and the IFS given by the follow-ing general map: ω : [0 , × T → T ( λ, x ) x + λ mod 1An interesting property of this IFS is that for any fixed x ∈ T we have ω ([0 , ×{ x } ) = T which implies that any sequence { x n } in T is a chain for the IFS. Soany δ -chain can be shadowed by itself by changing the sequence of parametersproving that it has shadowing property. On the other hand, each partial map is arotation and does not have shadowing property, thus if we fix a constant sequenceof parameters given ε >
0, for any δ > δ -pseudo-orbits, that are δ -chainswith constant sequence of parameters and cannot be shadowable by a chain withsequence of parameters.The following is an example of an expansive IFS of homeomorphisms with con-cordant shadowing property. Example 6.2.
Consider T = R / Z , a small ε > and Λ the closed ball in R centered in zero with radius ε . For x = ( x , x ) ∈ T consider the linear map f ( x , x ) = (2 x + x , x + x ) . The IFS will be given by the following general map: ω : Λ × T → T ( λ, x ) f ( x ) + λ mod 1Firstly, we observe that ω is the linear automorphism on the torus, which isexpansive.For a fixed σ = ( ..., λ − , λ − , λ , λ , ... ) ∈ Ω the structure of this IFS permits usto simplify the computation of the distance between the iterates of chains with thissequence of parameters. Let x ∈ X , the first iterate relative to σ is: ω λ ( x ) = f ( x ) + λ mod 1The second iterate for this sequence is: ω σ ( x ) = ω λ ( ω λ ( x ))= f ( f ( x ) + λ ) + λ mod 1= f ( x ) + f ( λ ) + λ mod 1= ω ( x ) + f ( λ ) + λ mod 1 N THE SPACE OF ITERATED FUNCTION SYSTEMS AND THEIR TOPOLOGICAL STABILITY7
By induction, we obtain for each positive k : ω σ k ( x ) = ω k ( x ) + k X i =1 f k − i ( λ i ) mod 1(1)Analogously, we find that: ω σ − k ( x ) = ω − k ( x ) − k X i =1 f − k + i ( λ − i ) mod 1(2)If we take y ∈ T , then for any k ∈ Z we have the following: d ( ω σ k ( x ) , ω σ k ( y )) = d ( ω k ( x ) , ω k ( y ))Since ω is expansive, so is the IFS.We can also prove that the IFS has concordant shadowing property, it comes fromthe fact that ω has shadowing property. Let ε >
0, then we have a δ > ω . To prove that the IFS has concordant shadowingproperty we take { y k } a δ -chain for a sequence σ = ( ..., λ − , λ − , λ , λ , ... ) ∈ Ω,then we can construct a sequence { x k } such that for any k > y k = x k − k X i =1 f k − i ( λ i ) mod 1(3) y − k = x − k + k X i =1 f − k + i ( λ − i ) mod 1(4)For k >
0, if we look for ω λ k +1 ( y k ), from the linearity of f we have: ω λ k +1 ( y k ) = f ( y k ) − λ k +1 mod 1= f ( x k ) − k X i =1 f k +1 − i ( λ i ) ! − λ k +1 mod 1(5) = ω ( x k ) − k +1 X i =1 f k +1 − i ( λ i ) mod 1Analogously we have: ω λ − k +1 ( y − k ) = ω ( x − k ) + k − X i =1 f − k + i − ( λ − i ) mod 1(6)On the other hand, using (3) and (4) we can explicit the expression of y k +1 and y − k +1 . So, from (5) and (6) we obtain that for any k ∈ Z : d ( x k +1 , ω ( x k )) = d ( y k +1 , ω λ k +1 ( y k )) < δ Thus, { x k } is a δ -pseudo orbit of ω and by the shadowing property there is z such that for any k ∈ Z we have: d ( x k , ω k ( z )) < ε From the expression obtained for ω σ k ( x ) in (1) and (2), we can conclude that: d ( y k , ω σ k ( z )) = d ( x k , ω k ( z )) < ε Then it is proved the concordant shadowing property.
A. ARBIETO, A. TRILLES
Remark.
Instead of the IFS be expansive, since ω is not positively expansive,the IFS also cannot be. 7. Proof of Theorem 1
A first observation is that the family of partial maps is uniformly equicontinuous,and it comes directly from the fact that the general maps is continuous.
Proposition 7.1.
The family of partial maps of an IFS is uniformly equicontinu-ous.Proof.
Let λ ∈ Λ. Since ω is continuous and Λ × X is compact, then ω is uniformlycontinuous which means that for any given ε > δ > d ( x, y ) = d (( λ, x ) , ( λ, y ) < δ implies that d ( ω ( λ, x ) , ω ( λ, y )) = d ( ω λ ( x ) , ω λ ( y )) < ε . (cid:3) With this fact, we can construct an auxiliary continuous function and using it,we show that every IFS is a compact subset of C ( X ) Proof of Theorem 1.
To see that K ∈ K ( C ( X )) is an IFS we just need to considerthe parameter space as K with the C -metric. The other continence requires alittle bit more.Let { ω λ : λ ∈ Λ } be an IFS, we want to show that it is a compact subset of C ( X ). For this, we define ϕ : Λ → C ( X ) given by ϕ ( λ ) = ω λ and we claim that ϕ is continuous.Let { λ n } be a sequence in Λ converging to λ . By the continuity of the firstvariable of ω : Λ × X → X , we obtain that { ω λ n } converges pointwise to ω λ .As { ω λ : λ ∈ Λ } is uniformly equicontinuous, so is { ω λ n } . Since X is compact,we obtain that the convergence is uniform, which implies convergence in the C -topology and consequently the continuity of ϕ . Therefore, ϕ (Λ) = { ω λ : λ ∈ Λ } iscompact and that completes the proof. (cid:3) Proof of Theorem 2
For some technical reasons sometimes we will need the phase space to be asmooth compact manifold, in those cases we will replace X by M and d will be aRiemannian metric on M , and every time we refer to a manifod we will be in thiscontext.Similarly to the definition of shadowing property, we can give another definitionwhich in advance seems to be weaker, permitting to obtain shadows only for finite δ -chains. Definition 8.1.
We say that an IFS ω has the finite shadowing property if for anygiven ε > , there exists δ > such that for any finite δ -chain { x , ..., x n } thereexists a chain { y k } such that d ( x k , y k ) < ε , for k = 1 , ..., n . As we are considering Λ and X compact, we have an equivalence between thesetwo definitions. Clearly shadowing property implies finite shadowing property andthe converse is given in the following lemma. Lemma 8.2.
If an IFS has finite shadowing property, then it has shadowing prop-erty.
N THE SPACE OF ITERATED FUNCTION SYSTEMS AND THEIR TOPOLOGICAL STABILITY9
Proof.
Let ε >
0, by hypothesis there exists δ > δ -chaincan be ε -shadowable. Let { x k } be a δ -chain, then for each natural number i thereexists y i ∈ X and σ i = ( λ i , λ i , ... ) ∈ Ω such that:(7) d ( ω σ ij ( y i ) , x j ) < ε, j = 0 , ..., i. By compactness of Λ and X , we can assume that { y i } converges to y and { λ ik } converges to λ k for all k . From this we can construct σ = ( λ , λ , ... ). Fixed n , wehave that ( λ i , ..., λ in ) converges to ( λ , ..., λ n ). As ω is continuous, ω σ in convergesto ω σ n and by (7) d ( ω σ n ( y ) , x n ) < ε . Thus, as n is arbitrary, we have that { ω σ k ( y ) } ε -shadows { x k } and the IFS has shadowing property. (cid:3) If we assume the dimension of the manifold to be at least 2, starting with theidentity map and making a local translation using a bump function we can commutepoints sufficiently close obtaining a diffeomorphism close to the identity in C -topology. In the uni-dimensional case a simple rotation can do this, so we have thefollowing lemma. Lemma 8.3.
For any given ε > , there exists δ > such that if x = y and d ( x, y ) < δ , then there exists a diffeomorphism f such that:(i) d C ( f, id ) < ε (ii) f ( x ) = y The following is the key lemma for the proof of Theorem 2. Having an IFS anda δ -chain we shall construct another IFS as close as wanted to the one we have. Wealso obtain a chain for this new IFS close to the initial δ -chain and such that theirsequences are δ -compatible, with this, the theorem becomes easy. Lemma 8.4.
Let ω be an IFS with a manifold M as phase space. Given ∆ > ,there exists δ > such that if { x , ..., x n } is a finite δ -chain with sequence σ , thenthere exists an IFS ˜ ω satisfying d H ( ω, ˜ ω ) < ∆ and a sequence ˜ σ such that ( σ, ˜ σ ) is ∆ -compatible satisfying d (˜ ω ˜ σ k ( x ) , x k ) < ∆ for k = 0 , ..., n .Proof. For ∆ >
0, from lemma 8.3 we obtain δ > d ( x, y ) < δ thenthere exists a diffeomorphism f such that:(i) d C ( f, id ) < ∆(ii) f ( x ) = y We assume 3 δ < ∆. Let { x , ..., x n } be a δ -chain. Since it is finite, using triangleinequality we can take { x = y , ..., y n } a 3 δ -chain such that:(i) d ( x k , y k ) < ∆, k = 0 , .., n (ii) ω λ k +1 ( y k ) = y k +1 , k = 0 , ..., n − k = 0 , ..., n − h k such that d C ( h k , id ) < ∆ and h k ( ω λ k +1 ( y k )) = y k +1 .We define ˜Λ = { , ..., n − }× Λ and ˜ ω : ˜Λ × M → M where ˜ ω ( k, λ, x ) := h k ◦ ω λ ( x ).As ω is continuous, so is ˜ ω and then it is a general map of an IFS. We claim thatfor each k ∈ { , ..., n − } and λ ∈ Λ we have d C (˜ ω ( k,λ ) , ω λ ) < ∆.Let x ∈ M , then we have:(8) d (˜ ω ( k,λ ) ( x ) , ω λ ( x )) = d ( h k ( ω λ ( x )) , ω λ ( x )) ≤ d C ( h k , id ) < ∆As x was arbitrary, we have d C (˜ ω ( k,λ ) , ω λ ) < ∆ and then d H ( ω, ˜ ω ) < ∆. We define ˜ σ = (˜ λ , ..., ˜ λ n ), where ˜ λ k := ( k, λ k ). Thus we have ˜ ω ˜ σ k ( x ) = y k andfrom (8) ( σ, ˜ σ ) is ∆-compatible. (cid:3) Lemma 8.5.
Every topologically stable IFS with a manifold as phase space hasfinite shadowing property. Moreover, if the sequence of parameters of the finite δ -chain has n elements, then they coincide with the firsts n elements of the sequenceof parameters of the shadow.Proof. Let ω : Λ × M → M be an IFS topologically stable with dimM ≥
2. Fora given ε >
0, from the definition of topological stability we obtain a ∆ > ω is an IFS with d H ( ω, ˜ ω ) < ∆, for each ( σ, ˜ σ ) ∆-compatible there exists acontinuous map h : M → M with the following properties:(i) d C ( ω σ k ◦ h, ˜ ω ˜ σ k ) < ε for all k ∈ N (ii) d C ( h, id ) < ε We assume ∆ < ε .Let δ < ∆6 and { x , ..., x n } a δ -chain with sequence σ , then from the previouslemma there exists ˜ ω with d H ( ω, ˜ ω ) < ∆, ˜ σ such that ( σ, ˜ σ ) is ∆-compatible and y such that d (˜ ω ˜ σ k ( y ) , x k ) < ∆ for k = 0 , ..., n .We will consider, by now, σ and ˜ σ infinite by complete with λ k = λ and˜ λ k = (1 , λ ) for k ≥ n . We remark that ( σ, ˜ σ ) is still δ -compatible.So, we obtain h : M → M a continuous map with the properties mentionedabove.We consider z = h ( y ), then { ω σ k ( z ) } is clearly a chain for ω and we observethat for k = 1 , ..., n : d ( x k , ω σ k ( z )) = d ( x k , ω σ k ( h ( y )) ≤ d ( x k , ˜ ω ˜ σ k ( y )) + d (˜ ω ˜ σ k ( y ) , ω σ k ( h ( y )) ≤ ∆ + ε < ε ε ε (cid:3) Corollary 8.6.
Every topologically stable IFS having a manifold as phase spacehas shadowing property. the sequence of parameters of the finite δ -chain has n elements, then they coincidewith the firsts n elements of the sequence of parameters of the shadow.Since we can shadow a finite δ -chain coinciding its n elements of sequence ofparameters with the firsts n elements of the shadow and M is compact we caneasily prove the Theorem 2. Proof of Theorem 2.
Let ε >
0. Let us consider δ > { x k } be a δ -chain with with sequence σ = ( λ , λ , ... ). For each n ∈ N if we consider { x , ..., x n } we obtain z n and σ n = ( λ n , λ n , ... ) such that forall 0 ≤ j ≤ n we have λ nj = λ j and consequently d ( x k , ω σ k ( z n )) = d ( x k , ω σ nk ( z n )) < ε (9) N THE SPACE OF ITERATED FUNCTION SYSTEMS AND THEIR TOPOLOGICAL STABILITY11
By compactness of M we can consider { z n } convergent and z its limit. Fixed k ∈ N , from (9) we have that d ( x k , ω σ k ( z )) < ε . So, { ω σ k ( z ) } is a chain that ε -shadows { x k } with the same sequence. (cid:3) Proof of Theorem 3
As we mentioned before, we shall use the existence of a unique chain shadowinga δ -chain with the same sequence of parameters. Definition 9.1.
We say that an IFS has uniqueness shadowing property if it hasconcordant shadowing property and there exists ε > such that for its respective δ from the concordant shadowing property we have that for any { x k } δ -chain withsequence σ there exists an unique y such that { ω σ k ( y ) } ε -shadows { x k } . Proposition 9.2. If ω is an η -expansive IFS and it has concordant shadowingproperty, then ω has the shadowing uniqueness property.Proof. Let ε < η . As ω has concordant shadowing property, we obtain δ > δ -chain is ε -shadowable by a chain with the same sequence. Let { x k } bea δ -chain with sequence σ , so there exists y such that { ω σ k ( y ) } ε -shadows { x k } .Now suppose there exists z such that { ω σ k ( z ) } ε -shadows { x k } , then we have: d ( y k , z k ) ≤ d ( y k , x k ) + d ( y k , z k ) < ε < η. Thus, as ω is η -expansive z = y and ω has shadowing uniqueness property. (cid:3) Lemma 9.3. If ω is an η -expansive IFS and σ = ( λ , λ , ... ) is a sequence, then forany given µ > there exists N > such that if x, y ∈ M and d ( ω σ n ( x ) , ω σ n ( y )) ≤ η for all n ≤ N , then d ( x, y ) < µ .Proof. Suppose that exists µ that fails the lemma. Then for each N ∈ N there exists x N and y N such that d ( ω σ k ( x N ) , ω σ k ( y N )) ≤ η for all k ≤ N but d ( x N , y N ) ≥ µ .So, we obtain { x N } N ∈ N and { y N } N ∈ N and by compactness we can assume theyare convergent, respectively to x and y . Now fixed n ∈ N , by continuity of theIFS we have that ω σ n ( x N ) converges to ω σ n ( x ) and ω σ n ( y N ) converges to ω σ n ( y ).As d ( ω σ n ( x N ) , ω σ n ( y N )) ≤ η for all n ≤ N , we obtain that d ( ω σ n ( x ) , ω σ n ( y )) ≤ η .On the other hand, d ( x N , y N ) ≥ µ for all N ∈ N wich implies d ( x, y ) ≥ µ . Thiscontradicts the hypothesis of ω be η -expansive. (cid:3) Proof of Theorem 3.
Let ε > ω be an IFS expansive with concordant shadowingproperty and η > ω . From the proposition 9.2we obtain that ω has shadowing uniqueness property, moreover from the proof weknow that any ε < η satisfies the shadowing uniqueness property, so let us consider ε < η , from the concordant shadowing property we have δ > δ -chain is uniquely ε -shadowable by a chain with the same sequence.Let ˜ ω be an IFS with d H ( ω, ˜ ω ) < δ . Fix σ ∈ Ω and x ∈ X . Let ˜ σ be a sequencesuch that ( σ, ˜ σ ) is δ -compatible.Since ( σ, ˜ σ ) is δ -compatible, we observe that { ˜ ω ˜ σ k ( x ) } is a δ -chain for ω withsequence σ , then there exists a unique point y x such that the chain { ω σ k ( y x ) } ε -shadows { ˜ ω ˜ σ k ( x ) } .We define h : X → X , by h ( x ) := y x and we observe that from the shadowinguniqueness property h is well defined and by construction d ( x, h ( x )) < ε for all x ∈ X . Thus, if h is continuous, then d C ( h, id ) < ε . We also observe that by construction d ( ω σ k ( h ( x )) , ˜ ω ˜ σ k ( x )) < ε for all k ∈ N and x ∈ M . So, if h iscontinuous, we also have d C ( ω σ k ◦ h, ˜ ω ˜ σ k ) < ε for all k ∈ N .We claim that h is continuous. Let µ > N ∈ N such that if x, y ∈ X and d ( ω σ k ( x ) , ω σ k ( y )) ≤ η , for all k ≤ N then d ( x, y ) < µ . For each k ∈ { , ..., N } , ω σ k and ˜ ω ˜ σ k are continuous, as M is compact,they are uniformly continuous and then for each k , there exists β k > β k > d ( x, y ) < β k , then d ( ω σ k ( x ) , ω σ k ( y )) < ǫ , and if d ( x, y ) < ˜ β k , then d (˜ ω ˜ σ k ( x ) , ˜ ω ˜ σ k ( y )) < ε . Take β = min { β k , ˜ β k : k = 0 , ..., N } . We observe that if d ( x, y ) < β then for k = 0 , ..., N we have: d ( ω σ k ( h ( x )) , ω σ k ( h ( y ))) ≤ d ( ω σ k ( h ( x )) , ˜ ω ˜ σ k ( x )) + d (˜ ω ˜ σ k ( x ) , ˜ ω ˜ σ k ( y ))+ d (˜ ω ˜ σ k ( y ) , ω σ k ( h ( y ))) < ε + ε + ε < η Thus, d ( x, y ) < β implies d ( ω σ k ( h ( x )) , ω σ k ( h ( y ))) < η for 0 , ..., N , which implies d ( h ( x ) , h ( y )) < µ and consequently h continuous and ω is topologically stable. (cid:3) Remark.
We remark that for this last theorem it is not required the phase spaceto be a manifold, it works for any compact metric space.
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