Iterated Function Systems : Transitivity and Minimality

We discuss iterated function systems generated by finitely many continuous self-maps on a compact metric space, with a focus on transitivity and minimality properties. More specifically, we are interested in topological transitivity, fiberwise transitivity, minimality and total minimality. A number of results that clarify the relations between topological transitivity and fiberwise transitivity are included. Furthermore, we generalize the notion of regular periodic decomposition for topologically transitive maps, introduced by John Banks [1], to iterated function systems. We will focus on the existence of periodic decompositions for topologically transitive iterated function systems. Finally, we show that each minimal abelian iterated function system consisting of homeomorphisms on a connected compact metric space is totally minimal.


Introduction
Transitivity and minimality have recently been the subject of considerable interest in topological dynamics [1,5,6,10,11,13,14].The concept of transitivity could trace back to Birkhoff [2,3].After that many articles dealt with such a topic.We continue this investigation in a context of iterated function systems.Iterated function systems are given by a (finite) collection of continuous maps on a metric space, that are composed for iterations.They have been studied extensively because of their role in the study of fractals [7,8].This paper studies two non equivalent notions of dynamical transitivity and discusses the relations between these notions, in the context of iterated function systems on compact metric spaces.
The other interesting issue we will be addressed to is the existence of a periodic decomposition.Regular periodic decompositions for topologically transitive maps introduced in [1].In fact, in a regular periodic decomposition, one can decompose the domain of a topologically transitive map into finitely many regular closed pieces with nowhere dense overlap in such a way that these pieces map into one another in a periodic fashion.Here, we generalize this concept to iterated function systems (IFSs) and provide some relevant results.Then we deal with minimal iterated function systems and minimal sets of IFSs.In particular, we show that each minimal abelian IFS on a compact connected metric space is totally minimal.
In order to state the main results, first we recall some standard definitions about iterated function systems.Next a brief description of the results will be given.

Preliminaries
We start collecting some basic concepts on iterated function systems.Let X be a compact metric space and F be a finite family of maps on X. Write F + for the semigroup generated by the collection F. Following [4,9], the action of the semigroup F + is called the iterated function system (or IFS) associated to F. In the rest of this paper, IFS(X; F) or IFS(F) will stand for an iterated function system generated by F. An iterated function system can be thought of as a finite collection of maps which can be applied successively in any order.Moreover, iterated function systems (IFSs) are a method of constructing fractals.
Throughout this paper, we assume that (X; d) is a compact metric space with at least two distinct points and without any isolated point.Also, we assume that IFS(F) is an iterated function system generated by a finite family F = {f 1 , . . ., f k } of continuous self-maps on a compact metric space X.
For the semigroup F + and x ∈ M the total forward orbit of x is defined by In a similar way, if the generators f i , i = 1, . . ., k, are injective then one can define the total backward orbit of x by Symbolic dynamic is a way to represent the elements of F + .We take Σ + k = {1, . . ., k} N endowed with the product topology.

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(2) IFS(F) is fiberwisely transitive if it admits an fiberwise dense orbit; (3) IFS(F) is forward minimal (or backward minimal ) if any point has a dense total forward orbit (or total backward orbit).
Theorem 1.1.Let IFS(F) be an iterated function system generated by finitely many homeomorphisms {f 1 , . . ., f k } defined on a compact metric space X.Then the following statements hold: 1) If IFS(F) has a ω-fiberwise dense orbit, for some sequence ω ∈ Σ + k , then the subset {x ∈ X : In particular, fiberwise transitivity implies topological transitivity; the converse is not true, in general.
2) If there exists h ∈ F + so that h admits a unique attracting fixed point x with dense total backward and total forward orbits then IFS(F) is fiberwisely transitive.
In [12], the authors proved that if the mappings f i , i = 1, . . ., k, preserve a finite measure with total support, then these two different notions of transitivity, topological transitivity and fiberwise transitivity, are equivalent.While we see that, in the non-conservative case, in general, the two properties are different.Theorem 1.1 above investigates their relationship.Let h ∈ F + .We say that the length of h is equal to j and denote it by |h| = j if h is a composition of j elements of the generating set F of IFS(F).For each iterated function system IFS(F) with generating set F we denote IFS(F n ) an iterated function system generated by Definition 1.1.We say that an iterated function IFS(F) is totally transitive (or totally minimal) if IFS(F n ) is transitive (or minimal) for all n ∈ N.
Let K(X) denote the set of non-empty closed subsets of X endowed with the Hausdorff metric topology.Then K(X) is also a complete metric space and it is compact whenever X is compact.For an iterated function system IFS(F) we define the associated Hutchinson operator by Following [4], we say that a compact set A is a self-similar set whenever In the following, we generalize the notion of a regular periodic decomposition for a topologically transitive map, introduced by John Banks [1], to iterated function systems.
Parham, Ghane and Ehsani called a periodic orbit of sets for the Hutchinson operator L. Clearly, the union of the X i is a self-similar set.
and the union of the X i is X.The number of sets in a decomposition will be called the length of the decomposition.A periodic decomposition is regular if all of its elements are regular closed.This means that for each 0 ≤ i ≤ n − 1, Cl(intX i ) = X i .In particular, X i ∩ X j is nowhere dense whenever i = j.Definition 1.4.Let IFS(F) be an iterated function system on a compact metric space X with generators The next theorem provides a necessary and sufficient condition for topological transitivity.
be a regular periodic decomposition for the iterated function system IFS(F) generated by a finite family F of open and continuous maps on a compact metric space X.Then IFS(F) is transitive if and only if IFS(F n ) is transitive on each X i .
Theorem 1.3.A minimal abelian iterated function system IFS(F) generated by finitely many homeomorphisms F = {f 1 , . . ., f k } defined on a connected compact metric space X is totally minimal and hence it is totally transitive.
The present paper is organized as follows.In Section 2, we discuss topological transitivity and fiberwise transitivity and clarify the relations between these two distinct notions.Moreover, the proof of Theorem 1.1 is also given in Section 2. In Section 3, we deal with the existence of regular periodic decompositions for topologically transitive iterated function systems, then we prove Theorem 1.2.In Section 4, we deal with minimal iterated function systems and minimal sets of IFSs and finally we prove Theorem 1.3.Finally, Section 5 of this paper outlines motivations and provides a discussion of the paper.

Transitivity of IFSs
Topological transitivity is a global characteristic of a dynamical system.This section will concentrate on transitivity properties of iterated function systems.
Throughout this section let F = {f 1 , . . ., f k } be a finite family of homeomorphisms defined on a compact metric space X and take IFS(F) the iterated function system generated by F. Assume that X has no any isolated point and write F + for the semigroup generated by F.
Here, we restrict ourselves to topological transitivity and fiberwise transitivity of IFSs.The first problem is, broadly speaking, a question about the relation between these two concepts of transitivity in the context of IFSs.

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In what follows, we will prove that fiberwise transitivity implies topological transitivity.However, the next example shows that these two properties are not equivalent.
Example 2.1.Let X be a a second countable Baire space and V = {V j : j ∈ N} a countable basis of X.Let us take a countable collection F = {f j : j ∈ N} consisting of self continuous maps defined on X, F + the semigroup generated by F and a point x ∈ X for which the following properties hold: (1) for each j, f j (x) ∈ V j ; (2) f j has an attracting fixed point x j in V j , moreover, V j is contained in the basin of attraction of x j ; (3) for all j = 1, f j has an attracting fixed point in V 1 .
Obviously, the total forward orbit of x is dense in X but x has not a fiberwise dense orbit.
Notice that if IFS(F) is a topologically transitive IFS on a second countable Baire space X, then the set of points with a dense total forward orbit is a residual set, for instance see [5,Prop. 1].In the next two results we generalize this observation to the fiberwisely transitive IFSs.Lemma 2.1.If IFS(F) has any point x such that for some ω ∈ Σ + k , x has ωfiberwise dense orbit, then the set of points with ω-fiberwise dense orbit is a residual subset of X.
Proof: Suppose that there exist a point x ∈ X and a sequence ω ∈ Σ + k such that the ω-fiberwise orbit of x, O + F (ω, x) = {f j ω (x) : j ∈ N}, is dense in X.Let us take a countable basis V = {V j : j ∈ N} of X and we set Since the fiberwise-orbit O + F (ω, x) of x is dense in X, hence there exist sequences {x j l } ⊂ O + F (ω, x) and {k n,l } ⊂ N, for which the following property holds: )) for all 0 ≤ i ≤ k n,l .This means that for each y ∈ B r n,l (x j l ), the segment orbit {y, f σ j l −1 ω (y), . . ., f ) is an open and dense subset.In particular, this proves that ∩ n∈N ∪ l∈N B r n,l (x j l ) is a residual subset contained in ∩ ∞ n=1 A n which consists of the points with ω-fiberwise dense orbit.✷ The next result relates forward and backward total orbits.

Parham, Ghane and Ehsani
Proposition 2.2.If there exists a point x ∈ X with ω-fiberwise dense orbit, for some ω ∈ Σ + k , then {y ∈ X : Proof: From the previous lemma, {y ∈ X : Cl(O + F (ω, y)) = X} is a residual subset of X.By assumption, the set {x, f ω 0 (x), . . ., f j ω (x), . . .} is dense in X.Let us take ) is dense, there exists a sequence {k n } of positive integers such that {x, . . ., f kn ω (x)} is 1 2ndense.As {x, . . ., f kn ω (x)} ⊂ O − (ω, x j ) for all j ≥ k n , we get that x j ∈ A n for each j ≥ k n .By continuity, one can find r j > 0 such that ) is an open neighborhood of x j and {x j : In particular, the next result can be followed immediately.Proof: Suppose that V = {V j : j ∈ N} is a countable basis of X and IFS(F) acts minimally on X.It is not hard to see that for each x ∈ X and every open subset V ⊂ X, there exists h ∈ F + so that h(x) ∈ V .Let α = (α 1 , . . ., α ℓ ) be a finite word so that h If we apply this conclusion for the countable basis V, we will provide a sequence Then for sequence ω, that is the concatenation of α (i) s, ω-fiberwise orbit of x is dense in X.This fact ensures that every point x has a dense fiberwise orbit and hence IFS(F) is fiberwisely transitive.✷ As a consequence of the above observations we get the next result.
The next result provides a sufficient condition for fiberwise transitivity.
Proposition 2.6.If there exists h ∈ F + with a unique attracting fixed point x so that the total backward and total forward orbits of x are dense in X, then there exists a sequence ω ∈ Σ + k so that the ω-fiberwise orbit of x is dense in X.In particular, IFS(F) is fiberwisely transitive.
Proof: Suppose that U and V are two open subsets of X.Since the total forward orbit of x is dense in X, there exist two mappings k 1 , g 1 ∈ F + such that g 1 (x) ∈ U and k 1 (x) ∈ V .By density of O − F (x), we can choose two sequences {z i } ⊂ U and {h i } ⊂ F + so that h i (z i ) = x and z i converges to g 1 (x) whenever i tends to infinity.On the other hand, by continuity of k 1 , one can find a small real r > 0 so that k 1 (B r (x)) ⊂ V , where B r (x) is the ball with radius r and center x.Let B be an open neighborhood of x so that h i (g 1 (x)) is contained in B for some large enough i.Since x is an attracting fixed point of h, one can find n ∈ N such that x ∈ h n (B) ⊂ B r (x), and hence k 1 (h n (h i (g 1 (x)))) ∈ V .Let us take g 1 (x) := x 1 and g 2 := k 1 • h n • h i , then g 1 (x) ∈ U and g 2 (x 1 ) ∈ V .We apply the above argument for a countable basis V = {V i : i ∈ N} of X.Then we will provide a sequence of mappings g i ∈ F + enjoying the following properties: g 1 (x) = x 1 , g i (x i ) = x i+1 and x i ∈ V i , for each i ∈ N. Let α i = (α 1,i , . . ., α ni,i ) be a finite word of the alphabets {1, . . ., k} so that k the concatenation of α i s then clearly the ω-fiberwise orbit of x is dense in X. ✷ Now, Theorem 1.1 is a consequence of Propositions 2.2 and 2.6.

Periodic decompositions of IFSs
This section will concentrate on dynamical properties relative to a regular periodic decomposition in the context of iterated function systems.We follow the approach introduced by Banks in [1] to our setting.We will prove that each iterated function system IFS(F) with a regular periodic decomposition D = {X 0 , . . ., X n−1 } of length n is transitive if and only if IFS(F n ) is transitive on each X i which establishes Theorem 1.2.
Throughout this section take F = {f 1 , . . ., f k } a finite family of continuous and open self maps defined on a compact metric space X and let IFS(F) be the iterated function system generated by F. Assume that X has no any isolated point and write F + for the semigroup generated by F. Lemma 3.1.For the iterated function system IFS(F) let D = {X 0 , X 1 , • • • , X n−1 } be a periodic orbit of subsets of X for the Hutchinson operator L. Then for each i = 0, . . ., n − 1, and each h ∈ F + with |h| = j, one has h(X i ) ⊆ X i+j (mod n) for all j ≥ 0. In particular, X i is an F n -invariant.
Proof: By induction for each j ≥ 0, L j (X i ) ⊆ X i+j , where This finishes the proof of the lemma.✷ Remark 3.2.Let L be the Hutchinson operator of an iterated function system IFS(F).Inductively, for each n ∈ N take L n = L • L n−1 .Then it is not hard to see that for each compact subset A ⊆ X one has which is a contradiction and the claim holds.
Notice that any two distinct forward minimal sets of an IFS are disjoint.Since X n is not disjoint from X 0 , we get that there exists ℓ ∈ N such that X ℓ = X 0 and X i ∩ X 0 = ∅ for all 0 < i < ℓ.This ℓ has to divide n, since otherwise, we would have X ℓ ⊂ X i for some 0 < i < ℓ, hence X n would be disjoint from X 0 .Now, the set Z = X 0 ∪ X 1 ∪ . . .∪ X ℓ−1 is a closed non-empty subset of X.We claim the it is forward F-invariant.In fact, for each i = 0, . . ., ℓ − 1, and Thus, by minimality of IFS(F), the set Z is equal to X.This completes the proof.✷ Since distinct minimal sets are pairwise disjoint, we get the next result by the previous theorem.
Corollary 4.4.Let IFS(F) be an abelian iterated function system on a compact metric space X.If X is connected and IFS(F) is forward minimal then it is totally minimal.Now, the proof of Theorem 1.3 is established.

Conclusion and discussion
Transitivity forms part of a popular definition of chaos in discrete dynamical systems.Two different transitivity properties, topological transitivity and fiberwise transitivity, for iterated function systems are investigated.The relation between these transitivities is studied.Summing up the main results obtained in Section 2, we have the following implifications: minimality ⇒ fiberwise transitivity ⇒ topological transitivity.
In general, topological transitivity is a weaker condition than fiberwise transitivity.Several conditions on spaces for topological transitivity to imply fiberwise transitivity are given.
It is known that the transitive systems are dynamically indecomposable.But they often admit a particular kind of decomposability in which their domains decompose into finitely many topologically non-trivial closed pieces which map into each other in a periodic fashion.The results of Section 3 give some information about regular periodic decompositions for transitive IFSs.As a consequence, it is shown that totally transitive iterated function systems do not have any regular periodic decomposition.Moreover, sufficient conditions for existence of a regular periodic decompositions for transitive IFSs are given.
In Section 4, minimal IFSs are studied.It is investigated that every minimal IFS on a compact and connected metric space is totally minimal.In this context, a natural question arises: Under which conditions a non-abelian minimal IFS on a compact and connected metric space is totaly minimal ?