Density of fiberwise orbits in minimal iterated function systems on the circle
aa r X i v : . [ m a t h . D S ] J u l DENSITY OF FIBERWISE ORBITS IN MINIMAL ITERATED FUNCTION SYSTEMSON THE CIRCLE
PABLO G. BARRIENTOS, ABBAS FAKHARI, AND A. SARIZADEHA bstract . We study the minimality of almost every orbital branch of minimal iterated functionsystems (IFSs). We prove that this kind of minimality holds for forward and backward minimalIFSs generated by orientation-preserving homeomorphisms of the circle. We provide newexamples of iterated functions systems where this behavior persists under perturbation of thegenerators.
1. I ntroduction
Group and semigroup actions on the circle is the main subject of recent studies and stillattract lots of attention. Much of these e ff orts have been focused so far to explore the richdynamics of finitely generated actions, i.e. dynamics generated by finitely many maps. Here,with regard to this issue, we consider dynamical systems generated by several maps on acompact metric space X, called iterated function systems (IFSs). More precisely, given maps f , . . . , f k of X , we study the action of the semigroup IFS( f , . . . , f k ) generated by these maps.The orbit of a point x for IFS( f , . . . , f k ) is a set of points y = h ( x ), for some h ∈ IFS( f , . . . , f k ).The IFS is minimal if every orbit is dense in X . A sequence of iterates x n + = f ω n ( x n ) with ω n ∈ { , . . . , k } chosen randomly and independently is called branch orbit starting at x = x .The sequence of compositions f n ω = f ω n ◦ · · · ◦ f ω , for every n ∈ N , is called orbital branch corresponding to ω = ω ω · · · ∈ Σ + k = { , . . . , k } N and O + ω ( x ) = { f n ω ( x ) : n ∈ N } , x ∈ X is the fiberwise orbit . Following [4] we consider any probability P on Σ + k with the followingproperty: there exists 0 < p ≤ / k so that ω n is selected randomly from { , . . . , k } in such away that the probability of ω n = i is greater or equal to p , for every i ∈ { , . . . , k } and every n ∈ N . More formally, in terms of conditional probability, P ( ω n = i | ω n − , . . . , ω ) ≥ p . Observe that the standard Bernoulli measures on Σ + k are typical examples of these kind ofprobabilities.In the recent works [4, 5, 6], the limit set of IFSs have been studied providing someconditions to guarantee minimality. In the present paper, focusing essentially on the same Key words and phrases.
Minimal IFS on the circle, minimality of orbital branches (fiberwise orbits). subject, we study the density of fiberwise orbits of minimal IFSs, mainly on the circle. Thisgoal is motivated in part by the main result in [4] saying that for a minimal IFS, almost everybranch orbit starting from an arbitrary point is dense in X . To be more precise, for eachpoint x there exists a set Ω ( x ) ⊂ Σ + k with P ( Ω ( x )) = O + ω ( x ) is dense in X , for every ω ∈ Ω ( x ). A logical question would be whether the set Ω ( x ) could be independent of thechoice of x . In other words, if almost every orbital branch of a minimal IFS acts minimally,i.e., X = O + ω ( x ) , for every x ∈ X and for almost every ω ∈ Σ + k . (1)We obtain an answer to this question when the ambient space is the circle.There are several obstacles and solutions to overcome the dependence of Ω ( x ) to the initialpoint x in general case. For instance, this is done in [3] under a condition named strongly-fibred . More precisely, the authors have shown that when a minimal IFS is strongly-fibredthen (1) holds for every ω ∈ Σ + k with dense orbit under the Bernoulli shift map. Recall that,a minimal IFS is “strongly-fibred” if for every open set U in X there exists ω ∈ Σ + k such that X ω ⊂ U , where X ω = ∞ \ n = ˆ f n ω ( X ) , ˆ f n ω = f ω ◦ · · · ◦ f ω n − ◦ f ω n . Historical examples of strongly-fibred minimal IFSs are those generated by contraction maps.In this case, there is a wealth of classical results, essentially going back to the seminal workof Hutchinson [13]. Meanwhile, a weakly hyperbolic
IFS is one for whichlim n →∞ diam( ˆ f n ω ( X )) = , for every ω ∈ Σ + k .It turns out that the minimal weakly-hyperbolic IFSs are strongly-fibred. Weakly hyperbolicIFS characterized in [2] as those for whichlim n →∞ sup ω ∈ Σ + k d ( f n ω ( x ) , f n ω ( y )) = x , y ∈ X . (2)A point q ∈ X is a (repelling / attracting) periodic point for IFS( f , . . . , f k ) if there exists h ∈ IFS( f , . . . , f k ) such that q is a (repelling / attracting) fixed point of h . Clearly, weakhyperbolicity prevents the existence of repelling periodic points. A weaker property than (2)which allows the existence of repelling periodic points for the IFS is “contraction of almostevery orbital branch” which was introduced as synchronization in [12]. This means that thereexists a set Ω ⊂ Σ + k with P ( Ω ) = ω ∈ Ω , there is a dense set W ( ω ) ⊂ X with lim n →∞ d ( f n ω ( x ) , f n ω ( y )) = , for every x , y ∈ W ( ω ) . Examples of minimal IFSs on the circle satisfying this kind of contracting property can befound in [14, Theorem 1] and [11, Theorem 1]. These examples assume, among other things,that the IFS is forward and backward minimal . That is, the semigroup generated by f , . . . , f k actsminimally on the circle and the same holds for the semigroup generated by their inverses.In our first result, we show the minimality of almost every orbital branch and the density ofperiodic points for these kind of IFSs. IBERWISE ORBITS IN MINIMAL IFSS 3
Theorem A.
Let f , . . . , f k be orientation-preserving homeomorphisms of the circle. Assume thatthe IFS generated by these maps is forward and backward minimal. Then there exists Ω ⊂ Σ + k with P ( Ω ) = such that S = O + ω ( x ) , for every x ∈ S and ω ∈ Ω .In particular, Ω contains all the sequences with dense orbit under the Bernoulli shift map and if thereexists a homeomorphism in IFS( f , . . . , f k ) which is not conjugate to a rotation, then the periodicperiodic points of IFS( f , . . . , f k ) are dense in S . Observe that the minimal IFSs mentioned in this theorem above are not strongly-fibred,as X ω = S , for every ω ∈ Σ + k . Thus, Theorem A provides a new condition under which aminimal IFS satisfies (1).While, there are a variety of recent concrete examples providing IFSs satisfying the as-sumption of Theorem A ([6, 14]), the theorem is in a way restrictive: only the circle isdiscussed. The main place in the proof where this assumption is needed is Antonov’s theo-rem stated on the circle. We try to overcome this limitation by providing examples of minimalIFSs satisfying (1) directly. Namely, we prove (1) in general case for the IFSs containing aminimal homeomorphism (see Proposition 1).Our next goal is to build IFSs on the circle satisfying the assumptions of Theorem A in arobust way. Here, the robustness is understood as the persistence of the minimality under C -perturbations of the generators of the initial IFS. There are various ways to construct suchIFSs. For instance, it is shown in [6] that every IFS generated by a pair of di ff eomorphisms C -close enough to rotations with no periodic points in common and without periodic ss -intervals (compact intervals whose endpoints are consecutive attracting periodic points ofdi ff erent generators) is forward and backward minimal in a robust way. However, the firstexamples of C -robustly forward and backward minimal IFSs on the circle going back to[9, 10]. These examples require that the IFS contains an irrational rotation (i.e., a rigid rotationof the circle with irrational rotation number) and a C -di ff eomorphism g with an attractinghyperbolic fixed point a with derivative Dg ( a ) lying in the interval (1 / , g . Theorem B.
Let g , g ∈ Di ff ( S ) be, respectively, an irrational rotation and an orientation-preserving di ff eomorphism which is not conjugate to a rotation. Then, there exists a C -neighborhood U of ( g , g ) such that the IFS generated by any pair ( f , f ) ∈ U is forward and backward minimal.Consequently, the following hold • minimality of almost every orbital branch : S = O + ω ( x ) , for every x ∈ S and ω ∈ Σ + with dense orbit under the Bernoulli shift map. • density of periodic points : the periodic points of IFS( f , f ) are dense in S . Moreover, ifthe IFS has a hyperbolic attracting / repelling periodic point then it has dense set of hyperbolicattracting / repelling periodic points. BARRIENTOS, FAKARI, AND A. SARIZADEH
An immediate consequence is that every IFS with an irrational rotation can be approxi-mated by C -robustly forward and backward minimal one. If the IFS contains a di ff eomor-phism which is conjugate to an irrational rotation, one can conjugate the IFS to one containingthis irrational rotation. Unluckily, we cannot use the previous theorem since the conjugacymap is not di ff erentiable in general. However, the following result shows that, even in thiscase, the IFS can be approximated by a C -robustly forward and backward minimal one.The main tool applied here is the smooth conjugacy for the circle di ff eomorphisms providedin [7]. Corollary C.
Let g , . . . , g k be C -di ff eomorphisms of the circle with k ≥ and suppose that thereis a map in IFS( g , . . . , g k ) with irrational rotation number. Then for every C -neighborhood U of ( g , . . . , g k ) there is ( f , . . . , f k ) ∈ U such that the IFS generated by these maps is C -robustly forwardand backward minimal.Proof. Let f be the C -di ff eomorphisms in IFS( f , . . . , f k ) with irrational rotation number α .According to [7], there exists a sequence of C -di ff eomorphisms h n such that h n ◦ f ◦ h − n tends to the rotation R α in the C -topology as n → ∞ . Thus, F n = h n ◦ IFS( f , . . . , f k ) ◦ h − n can be taken arbitrarily C -close to an IFS containing R α . Then, if necessary, by means of asmall perturbation of the generators f , . . . , f k we get a semigroup G n arbitrarily close to F n satisfying Theorem B. Consequently, the IFS given by h − n ◦ G n ◦ h n is C -robustly forwardand backward minimal and arbitrarily close to IFS( f , . . . , f k ). This concludes the proof ofthe result. (cid:3) Let us end this introduction by asking if every minimal IFS of C -di ff eomorphisms, evenon the circle, can be approximated by C -robustly minimal one.The proof of Theorem A is handled in Secion 2. Section 3 is devoted to the proof ofTheorem B and some auxiliary lemmas.2. M inimality of orbital branches : P roof of T heorem AWe first study the minimality of almost every orbital branch in the special case that the IFScontains a minimal homeomorphism. To do this, we need a bit of notation. Given a finiteword σ = σ . . . σ n in the alphabet { , . . . , k } we denote by | σ | the length of σ . Proposition 1.
Let f , . . . , f k be homeomorphisms of a compact metric space X. Assume that IFS( f , . . . , f k ) contains a minimal homeomorphism. Then there exists Ω ⊂ Σ + k with P ( Ω ) = suchthat X = O + ω ( x ) , for every x ∈ X and ω ∈ Ω .Moreover, Ω contains all the sequences with dense orbit under the shift map.Proof. By the compactness of X , there is a countable open base B . Let B be an element of B and fix x ∈ X . IBERWISE ORBITS IN MINIMAL IFSS 5
Claim 1.
There is Ω ( B ) ⊂ Σ + k , with P ( Ω ( B )) = , such that for every ω ∈ Ω ( B ) ,f n ω ( x ) ∈ B , for some n ≥ . (3) Moreover, Ω ( B ) contains all the sequences with dense orbit under the shift map. By doing this, the proof of the proposition can be derived from the countability of B .Indeed, it follows that Ω = \ B ∈B Ω ( B )has full P -measure, contains all the sequences with dense orbit under the shift map and itholds that for every ω ∈ Ω and B ∈ B , O + ω ( x ) ∩ B , ∅ for every x ∈ X .This completes the proof of the proposition. (cid:3) Proof of Claim 1.
We first provide ℓ ∈ N and a word σ with | σ | = ℓ in such a way that for every ω ∈ C σ and x ∈ X , f t ω ( x ) ∈ B for some 0 < t ≤ ℓ , (4)where C σ denotes the cylinder in Σ + k around the finite word σ = σ . . . σ ℓ . Let h = f α s ◦· · ·◦ f α bethe minimal homeomorphism in IFS( f , . . . , f k ). Observe that the minimality h is equivalentto that of h − , so X = r [ i = h − i ( B ) for some r ∈ N .Hence, for each x ∈ X , there is i ∈ { , . . . , r } with h i ( x ) ∈ B . Put ℓ = rs . Thus, defining σ = α . . . α where α = α . . . α s appears r -times one can conclude (4).Continuing the proof of Claim 1, let Γ ( B ) be the set of elements ω ∈ Σ + k for which thereexists x ∈ X such that f j ω ( x ) < B , for all j ∈ N . Observe that Γ ( B ) ⊂ ∞ \ n = Γ ( B , n ) , (5)where Γ ( B , n ) = { ω : ∃ x ∈ X such that f j ω ( x ) < B for every j ≤ n + ℓ } . Equations (4) and (5)imply that P ( Γ ( B )) ≤ P ( Γ ( B , n )) ≤ (1 − p ℓ ) · P ( Γ ( B , n − ℓ )) ≤ (1 − p ℓ ) + [ n /ℓ ] → n → ∞ . Thus, taking Ω ( B ) = Σ + k \ Γ ( B ), one gets P ( Ω ( B )) = Ω ( B ). This concludes the proofof Claim 1. (cid:3) The next theorem establishing the synchronization phenomenon is the key ingredient toprove Theorem A. We denote by ν any Bernoulli measure on Σ + k . BARRIENTOS, FAKARI, AND A. SARIZADEH
Theorem 2.1 (Antonov [1]) . Let f , . . . , f k be orientation-preserving homeomorphisms of the circlesuch that the IFS generated by them is forward and backward minimal. Then exactly one of thefollowing statements holds:(1) there exists a common invariant measure of all the f i ’s, and all these maps are simultaneouslyconjugate to rotations;(2) for any two points x , y ∈ S , there exists Ω = Ω ( x , y ) ⊂ Σ + k with ν ( Ω ) = such that lim n →∞ d ( f n ω ( x ) , f n ω ( y )) = , for every ω ∈ Ω ; (3) there exists an integer ℓ > and an order ℓ orientation-preserving homeomorphism ϕ , suchthat it commutes with all the f i ’s, and after passing to the quotient circle S / ( ϕ i ( x ) ∼ ϕ j ( x )) the conclusion of (2) are satisfied, for the new maps g i . The next lemma which is essentially consequence of Antonov’s theorem plays a key rolein proving Theorem A. Our arguments rely upon the same basic strategy as those of [8].
Lemma 2.2.
Under the assumptions of Theorem 2.1, if there is a homeomorphism in
IFS( f , . . . , f k ) which is not conjugate to a rotation then there exists an integer ℓ ≥ such that for ν -almost every ω ∈ Σ + k there are points r i = r i ( ω ) ∈ S , for i = , . . . , ℓ , such that lim n →∞ d ( f n ω ( x ) , f n ω ( y )) = , for every x , y ∈ U.where U is any connected component of S \ { r , . . . , r ℓ } .Proof. Assume that we are in the case (2) of Theorem 2.1. For any arc I ⊂ S there is a subset Ω ( I ) ⊂ Σ + k with ν ( Ω ( I )) = f n ω ( I ) or its complement f n ω ( S \ I )tends to 0 for every ω ∈ Ω ( I ). Consider a sequences of finer partitions P m of the circle intoclosed arcs with rational endpoints whose length goes to zero. Since we have a numerablenumber of the arcs arcs Ω = \ m ∈ N \ I ∈P m Ω ( I ) ⊂ Σ + k satisfies ν ( Ω ) =
1. For each ω ∈ Ω , as the total length of the circle is preserved, there is exactlyone arc I m = I m ( ω ) of the partition P m such that the length of f n ω ( I m ) tends to 1 and the length ofthe images of its complement tends to 0. These closed arcs I m , m ∈ N form a nested sequencewhose length goes 0, and so there exists a unique intersection point r ( ω ) ∈ S . Therefore, forevery two points x , y ∈ S \ { r ( ω ) } , here is a natural number m such that x , y ∈ S \ I m , andhence lim n →∞ d ( f n ω ( x ) , f n ω ( y )) = . Assume now that we are in the case (3). Passing to the quotient space S /ϕ and arguingas above for quotient dynamics g , . . . , g k , the same behavior follows. That is, there exists Ω ⊂ Σ + k with ν ( Ω ) = ω ∈ Ω there is a point r = r ( ω ) ∈ S /ϕ withthe property that the length of any arc which does not contain r tends to 0 by iteration ofthe orbital branch g n ω . Since ϕ is an order ℓ > IBERWISE ORBITS IN MINIMAL IFSS 7 the equivalence class r has exactly ℓ di ff erent representative r i = r i ( ω ) ∈ S , for i = , . . . , ℓ .Moreover, ϕ i ( r ) = r i , for i = , . . . , ℓ .Let U be a connected component of S \ { r , . . . , r ℓ } . Consider x , y ∈ U and denote by [ x , y ]the arc with endpoint x and y containing in U . Observe that r < π ([ x , y ]) where π : S → S /ϕ is the projection on the quotient space. Hence, the length of this arc π ([ x , y ]) in the S /ϕ goesto zero by iteration g n ω . This means, in particular, thatlim n →∞ d ( f n ω ( x ) , f n ω ( y )) = . This concludes the proof of the lemma. (cid:3)
Now, we will prove Theorem A. Denoting by | I | the length of an arc I ⊂ S . Proof of Theorem A.
Suppose that we are in the case (1) of Antonov’s theorem. Then, there is aminimal homeomorphism among the generator f , . . . , f k of the IFS. Otherwise, the f i ’s beingsimultaneously conjugate to rational rotations. Hence, the group generated by f , . . . , f k isfinite and it cannot act minimally on the circle. Proposition 1 implies the minimality ofalmost every orbital branch in this case.Now, assume that there exists a homeomorphism in IFS( f , . . . , f k ) which is not conjugateto a rotation. By Lemma 2.2, there exists an integer ℓ ≥ Σ ⊂ Σ + k with ν ( Σ ) = ω ∈ Σ there are points r i = r i ( ω ) ∈ S , for i = , . . . , ℓ , such that for anyconnected component U of S \ { r , . . . , r ℓ } ,lim n →∞ d ( f n ω ( x ) , f n ω ( y )) = , for every x , y ∈ U . (6)Let ϑ ∈ Σ + k has dense orbit under the Bernoulli shift map and consider a point x and an openset I in S . We want to see that O + ϑ ( x ) ∩ I , ∅ . Claim 2.
There exists a finite word τ such that for each z ∈ S , there is ≤ t = t ( z ) ≤ | τ | with theproperty that f τ t ◦ · · · ◦ f τ ( z ) ∈ I. By doing this, using the fact that the orbit of ϑ is dense under the Bernoulli shift map σ ,one can choose n ≥ σ n ( ϑ )] i = τ i , for i = , . . . , | τ | . Taking z = f n ϑ ( x ) one has that f n + t ( z ) ϑ ( x ) ∈ I . which proves the minimality of the orbital branch along ϑ . Since the set of all elements of Σ + k with dense orbit has full P -probability, one can conclude the minimality of almost everyorbital branch. So, the proof of Theorem A is done in the first part. Proof of Claim 2.
Let D be a countable dense subset of S . By [4], there is Ω ( q ) ⊂ Σ + k with ν ( Ω ( q )) = S = O + ω ( q ), for every ω ∈ Ω ( q ). Consider Ω = \ q ∈ D Ω ( q ) ∩ Σ . BARRIENTOS, FAKARI, AND A. SARIZADEH
Observe that ν ( Ω ) = O + ω ( q ) is dense in S , for every q ∈ D and every ω ∈ Ω . Fix ω ∈ Ω .By (6), for every ε >
0, there exists K = K ( ω, ε ) ∈ N such that | f n ω ( U i ) | < ε, for every n ≥ K and i = , . . . , ℓ, (7)where the arcs U i = U i ( ω, ε ) ⊂ S are the connected components of the set S \ ( B ε ( r ) ∪ · · · ∪ B ε ( r ℓ )) . Here, B r ( x ) denotes the open ball of radius r > x .In view of the backward minimality, one can recursively find maps h , . . . , h ℓ in IFS( f , . . . , f k )such that h − i ◦ · · · ◦ h − ( r i ) ∈ I , for every i = , . . . , ℓ . Now, consider ε > h − i · · · ◦ h − ( B ε ( r i )) ⊂ I , for every i = , . . . , ℓ .Equation (7) implies that by iteration of f n ω the arcs U i are contracted. Since each of thesearcs has points of D and ω ∈ Ω , one can move these arcs around the circle to any openset. In particular, there exist non-negative integer numbers n = n ( ω ) , . . . , n ℓ = n ℓ ( ω ) with n ≥ K ( ω, ε ) so that f n + ··· + n i ω ( U i ) ⊂ I , for every i = , . . . , ℓ .Let γ i be the words corresponding to the maps h i for i = , . . . , ℓ . Put τ = γ ℓ . . . γ ω . . . ω n + ··· + n ℓ . The word τ is the desired one in Claim 2. To show this it is su ffi cient to show that for any z ∈ S there is 0 ≤ t ≤ | τ | such that f τ t ◦ · · · ◦ f τ ( z ) ∈ I . Consider the point y = h ◦ · · · ◦ h ℓ ( z ) = f τ s ◦ · · · ◦ f τ ( z ) , where s = | γ | + · · · + | γ ℓ | . We have three cases: • If y ∈ U ∪ · · · ∪ U ℓ , then f n + ··· + n i ω ( y ) ∈ I , for some i ∈ { , . . . , ℓ } . • If y ∈ B ε ( r i ) with i ∈ { , . . . , ℓ − } , then h i + ◦ · · · ◦ h ℓ ( z ) = h − i ◦ · · · ◦ h − ( y ) ∈ h − i ◦ · · · ◦ h − ( B ε ( r i )) ⊂ I . • If y ∈ B ε ( r ℓ ), then z = h − ℓ ◦ · · · ◦ h − ( y ) ∈ h − ℓ ◦ · · · ◦ h − ( B ε ( r ℓ )) ⊂ I .In any case, one has that f σ t ◦ · · · ◦ f σ ( z ) ∈ I for some 0 ≤ t ≤ | σ | as we want. (cid:3) Continuing the proof of Theorem A, we focus on the second part, which is the density ofperiodic points. Let J ⊂ S be any arbitrary small open interval. We will show that thereexists a periodic point of IFS( f , . . . , f k ) in J . Claim 3.
There is a fixed point a of a map g in
IFS( f , . . . , f k ) such that at least for one-side a becomesas an attracting point of g.Proof. We use the notation applied in the proof of the first part. By (7) and the density of D in S , for every ω ∈ Ω and every i ∈ { , . . . , ℓ } , there is a su ffi ciently large number n such that f n ω ( U i ) ⊂ U i and | f n ω ( U i ) | < | U i | . By Brouwer’s fixed-point theorem, the mapping g = f n ω has afixed point a in U i satisfying the required attracting property. (cid:3) IBERWISE ORBITS IN MINIMAL IFSS 9
Forward minimality allows us to find F ∈ IFS( f , . . . , f k ) such that F ( J ) ∩ B has non-emptyinterior where B is the basin of attraction of a for g . Let V ⊂ J be a non-degenerated closedarc such that F ( V ) ⊂ B . Again by the forward minimality of the IFS, there is G ∈ IFS( f , . . . , f k )such that G ( a ) belongs to the interior of V . By the continuity of G , there is δ > G (( a − δ, a + δ )) is contained in V . Now, since F ( V ) is contained in the basin of a , there is m ≥ g m ◦ F ( V ) ⊂ ( a − δ, a + δ ) and so, G ◦ g m ◦ F ( V ) ⊂ V . By Brouwer’s fixed-pointtheorem, G ◦ g m ◦ F has a fixed point in V ⊂ J . This implies the desired density of periodicpoints of IFS( f , . . . , f k ) and eventually ends the proof of the second part of Theorem A. (cid:3) Remark 1.
The proof of the second part of Theorem A can be improved provided thegenerators f , . . . , f k are C -di ff eomorphisms and there is a hyperbolic attracting fixed pointof some map in IFS( f , . . . , f k ). This improvement involves taking a large iteration g m in sucha way that G ◦ g m ◦ F is contracting on V . After that, one uses Banach’s fixed-point theoremconcluding the existence of a unique hyperbolic attracting fixed point of G ◦ g m ◦ F in V .In fact, we obtain something more which is the density of hyperbolic attracting periodicpoints of the IFS. Similar argument also runs to the backward IFS getting the same result forrepellers. 3. R obust minimal IFS s : P roof of T heorem BLet g , g ∈ Di ff ( S ) be, respectively, the irrational rotation x → x + α and an orientation-preserving di ff eomorphism which is not conjugate to a rotation. This implies that IFS( g , g )is not conjugate to a semigroup of rotations. As we have shown in the proof of Theorem A,in this case, there exists g ∈ IFS( g , g ) with a fixed point a such that at least for one-side a becomes an attracting point of g . Without lose of generality, we suppose that g = g and A = ( a , a + ε ) is its local basin of attraction, i.e., 0 < Dg ( x ) <
1, for every x ∈ A . Consider δ > | δ − ε | > | a − g ( a + ε ) | and set B = (cid:16) g ( a + ε ) , g ( a + ε ) (cid:17) and D = (cid:16) a , g ( a + ε ) (cid:17) . (8)Observe that since g is an irrational rotation, there exist n , . . . , n k ∈ N such that for h i = g n i ◦ g it holds(1) B ⊂ h ( B ) ∪ · · · ∪ h k ( B ),(2) g n i ( D ) ⊂ ( a + δ, a + ε ), for every i = , . . . , k ,(3) there is λ < Dh i ( x ) < λ , for every x ∈ ( a + δ, a + ε ) and i = , . . . , k .The third term holds by the equality Dg n i = g , one can cover the whole circle by finitely many iterations of B .That is, there exist times m , . . . , m s and ˜ m , . . . , ˜ m r such that for T i = g m i and S i = g ˜ m i it holds S = s [ i = T i ( B ) = r [ i = S − i ( B ) . (9)Now, we shall prove that conditions (1), (2), (3) and (9) imply that the IFS generated by g and g is C -robustly minimal. Namely, we want to prove that for every Φ = ( f , f ) close enough to ( g , g ) in the C -topology, S = Orb +Φ ( x ) , for every x ∈ S where, as before, Orb +Φ ( x ) = { h ( x ) : h ∈ IFS( f , . . . , f k ) } .First of all, observe that (1), (2), (3) and (9) are C -robust conditions i.e., there are C -neighborhoods V i of g i , for i = ,
2, such that for each ( f , f ) ∈ V × V there are maps˜ h , . . . , ˜ h k , ˜ T , . . . , ˜ T m , ˜ S , . . . , ˜ S r in IFS( f , f ) so that conditions (1), (2), (3) and (9) hold for thesemap for the same B and D . Hence, to prove the robust minimality of the IFS generated by g and g , it su ffi ces to show that (1), (2), (3) together with (9) imply the forward minimality.We begin by a simple observation that by (8) and (2), for every i . . . i n , with i j ∈ { , . . . , k } ,it holds h i ◦ · · · ◦ h i n ( B ) ⊂ ( a + δ, a + ε ) . (10)The rest of the proof will be done through a few steps for which we need a bit of notation.For any n >
1, define recursively sets B ni ... i n = h i n ( B n − i ... i n − ) = h i n ◦ · · · ◦ h i ( B ) , for i j = , . . . , k and j = , . . . , n . The proof of the lemma below is short and elementary yetreleases us from some notation inconsistency in this context. Lemma 3.1.
For every n ∈ N it holdsB ni ... i n + ⊂ k [ i = B n + i i ... i n + and B ⊂ k [ i ,..., i n + = B n + i ... i n + . Proof.
The proof is by induction on n . First, we show that B i ⊂ k [ i = B i i and B ⊂ k [ i , i = B i i . By condition (1), B ⊂ B ∪ · · · ∪ B k , and so, k [ i = B i i = k [ i = h i ( B i ) = h i ( k [ i = B i ) ⊃ h i ( B ) = B i . From this one gets that k [ i , i = B i i ⊃ k [ i = B i ⊃ B . Now, we assume that the lemma holds for n − n . In the same wayas before, k [ i = B n + i ... i n + = k [ i = h i n + ( B ni ... i n ) = h i n + ( k [ i = B ni ... i n ) . IBERWISE ORBITS IN MINIMAL IFSS 11
By hypothesis of the induction, one has that B n − i ... i n ⊂ ∪ ki = B ni i ... i n and so k [ i = B n + i ... i n + ⊃ h i n + ( B n − i ... i n ) = B ni ... i n + . For the second inclusion of the lemma, we first note that for every 1 ≤ ℓ ≤ n and every i j = , . . . , k with j = , . . . , ℓ + B ℓ i ... i ℓ + ⊂ k [ i = B ℓ + i i ... i ℓ + . Hence, k [ i ,..., i n + = B n + i ... i n + ⊃ k [ i ,..., i n + = B ni ... i n + ⊃ · · · ⊃ k [ i n + = B i n + ⊃ B , and the proof of the lemma is completed. (cid:3) Lemma 3.2.
Under the conditions (1) , (2) and (3) , for every x ∈ B there is a sequence ( i j ) j > ,i j ∈ { , . . . , k } such that x = lim n →∞ h i ◦ · · · ◦ h i n ( y ) , for every y ∈ B . Proof.
Since B ⊂ B ∪ · · · ∪ B k , for every x ∈ B there is i ∈ { , . . . , k } such that x ∈ B i . We nowproceed recursively. For n >
1, suppose that i j ∈ { , . . . , k } , for j = , . . . , n , has chosen in sucha way that x ∈ B ni n ... i . Using the inclusion B ni n ... i ⊂ ∪ kj = B n + ji n ... i , given by Lemma 3.1, one canchoose i n + ∈ { , . . . , k } such that x ∈ B n + i n + i n ... i . From this, a sequence i = i i . . . = ( i j ) j > ofpositive integers can be constructed so that x ∈ B ni n ... i , for every n ≥
1. Hence, one gets x ∈ \ n ≥ B ni n ... i = \ n ≥ h i ◦ · · · ◦ h i n ( B ) = \ n ≥ A n , where A n = ∩ n ℓ = h i ◦ · · · ◦ h i ℓ ( B ), for every n ∈ N .Since A n + ⊂ A n ⊂ h i ◦ · · · ◦ h i n ( B ), using the mean-value theorem, there are z j ∈ B , for j = , . . . , n , such thatdiam( A n ) ≤ diam( h i ◦ · · · ◦ h i n ( B )) ≤ n − Y j = Dh i j ( h i j + ◦ · · · ◦ h i n ( z j )) · Dh i n ( z n ) · diam( B ) . According to (10), h i j + ◦ · · · ◦ h i n ( z j ) ∈ ( a + δ, a + ε ) and thus, condition (3) implies thatdiam( A n ) ≤ λ n diam( B ). Consequently, A n being a nested sequence of sets whose diametersgoes to zero and hence one gets { x } = \ n ≥ B ni n ... i = \ n ≥ h i ◦ · · · ◦ h i n ( B ) . This implies that for every y ∈ B and every natural number n , | h i ◦ · · · ◦ h i n ( y ) − x | ≤ diam( h i ◦ · · · ◦ h i n ( B )) ≤ λ n diam( B ) . This completes the proof of the lemma. (cid:3)
Proof of Theorem B.
Now, we show the minimality of any IFS generated by two di ff eomor-phisms g , g satisfying conditions (1), (2), (3) and (9). Fix a point x ∈ S and an open set I ⊂ S . The previous lemma implies that B ⊂ Orb +Φ ( x ) , for every x ∈ B .By (9), we find i ∈ { , . . . , r } such that S i ( x ) ∈ B . Similarly, B ∩ T − j ( I ) contains an open set forsome j ∈ { , . . . , s } . Using the density of the orbits in B , one can find h ∈ IFS( g , g ) such that h ◦ S i ( x ) ∈ T − j ( I ). Thus, T j ◦ h ◦ S i ( x ) ∈ I . This shows that IFS( g , g ) is minimal as we want.In order to show the robust backward minimality, we observe that we can obtain (1), (2), (3)and (9) but now for IFS( g − , g − ). Thus, applying the same argument for IFS( g − , g − ), weconclude the robust minimality of this system. Finally, to conclude Theorem B, it su ffi ces toapply Theorem A and Remark 1. (cid:3) A cknowledgments We are grateful to Lorenzo J. D´ıaz for his assistance and review of preliminary versionsof this paper and Ale Jan Homburg and Malicet Dominique for their comments. During thepreparation of this article the authors were partially supported by the following fellowships:P. G. Barrientos by MTM2011-22956 project (Spain) and CNPq post-doctoral fellowship(Brazil); A. Fakhari by grant from IPM. (No. 91370038). A. Fakhari and A. Sarizadeh thankICTP for their hospitality during their visit.R eferences [1] V. A. Antonov.
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