# Robust Density of Periodic Orbits for Skew Products with High Dimensional Fiber

Authors: Fatemeh Helen Ghane; Mahboubeh Nazari; Mohsen Saleh; Zahra Shabani
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider step and soft skew products over the Bernoulli shift which have anm-dimensional closed manifold as a fiber. It is assumed that the fiber maps Hölder continuously depend on a point in the base. We prove that, in the space of skew product maps with this property, there exists an open domain such that maps from this open domain have dense sets of periodic points that are attracting and repelling along the fiber. Moreover, robust properties of invariant sets of diffeomorphisms, including the
coexistence of dense sets of periodic points with different indices, are obtained.


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## Body

## 1. Introduction

In [1], Gorodetski and Ilyashenko studied certain properties of skew product maps over the Bernoulli shift and the Smale-Williams solenoid, with a fiber S1. They provided an open set in the space of these skew products such that each mapping from this open set has a dense set of periodic orbits that are attracting and repelling along the fiber.In this paper, we improve their results to skew product maps which have anm-dimensional closed manifold M as a fiber. Moreover, we prove that small perturbations of these skew products in the space of all diffeomorphisms have partially hyperbolic invariant sets. Also, they admit dense subsets of periodic points with different indices.To be more precise, let us describe skew product maps which apply here in detail.From now on, the ambient fiber spaceM will be an m-dimensional closed manifold and its metric is geodesic distance and the measure is the Riemannian volume.Consider diffeomorphismsfi, i=1,…,k, defined on M. The iterated function system  ℱ(M;f1,…,fk) is the semigroup generated by f1,…,fk, that is, the set of all maps ftj∘⋯∘ft1, where tj,…,t1∈{1,…,k}.Theℱ-orbit of x∈M is the set of points fti∘⋯∘ft1(x), tj≥0.An iterated function systemF(M;f1,…,fk) is called minimal if each closed subset A with fi(A)⊂A, for all i, is empty or coincides with M. This means that ℱ-orbit of each x∈M is dense in M.Letfi, i=0,1, be diffeomorphisms of M. A step skew product over the Bernoulli shift σ:Σ2→Σ2 is defined by
(1)F:Σ2×M⟶Σ2×M;(ω,x)⟶(σω,fw0(x)),
where Σ2 is the space of two-sided sequences of 2 symbols {0,1}. Consider the following standard metric on Σ2:
(2)d(ω,ω′)=2-n,
where n=min{|k|;ωk≠ωk′} and ω,ω′∈Σ2.Let us note that an iterated function system can be embedded in a single dynamical system, the skew productF of the form (1), such that the action orbits of the iterated function system ℱ with generators fi coincide with the projections of positive semitrajectories of the skew product F onto the fiber along the base.Asoft skew product over the Bernoulli shift is a map
(3)G:Σ2×M⟶Σ2×M;(ω,x)⟶(σω,gω(x)),
where the fiber maps gω are diffeomorphisms of the fiber into itself.We would like to mention that in contrast to step skew products, the fiber maps of soft skew products depend on the whole sequenceω.Skew products play an important role in the theory of dynamical systems. Many properties observed for these products appear to persist as properties of diffeomorphisms [1, 2].Letw be a finite segment on the alphabets {0,1}. We denote by {⋯∣w⋯} an arbitrary infinite sequence ω in which w occurs starting from the zeroth position. In a similar way, we introduce the notation {⋯w∣⋯} and {⋯w∣w′⋯}. We also denote by |w| the length of w.We recall that a mapF is called topologically mixing if for each nonempty open sets U,V∈Σ2×M,Fn(U) intersects with V for all large enough n∈ℕ.For a diffeomorphismf of M, a compact f-invariant set Λ has a dominated splitting if
(4)TΛM=E1⊕⋯⊕Ek,
where each Ei is nontrivial and Df-invariant for 1≤i≤k and there exists an m∈ℕ such that
(5)∥Dfn∣Ei(x)∥∥(Dfn∣Ej(x))-1∥≤12,
for every n≥m, i>j and x∈Λ.The setΛ is partially hyperbolic if it has a dominated splitting
(6)TΛM=E1⊕⋯⊕Ek
and there exists some n∈ℕ such that Dfn either uniformly contracts E1 or uniformly expands Ek.We are now ready to state our main results. The first result describes the robust density of attracting and repelling periodic orbits along the fiber.Theorem 1.
There existC1 diffeomorphisms fi:M→M, i=0,1, and C1-neighborhoods U0(f0), U1(f1)⊂Diff1(M) such that for any g0∈U0 and g1∈U1, the periodic orbits of the step skew product F of the form (1) with the fiber maps gi,i=0,1, which are attracting (or repelling) along M, are dense in Σ2×M.By applying the Hölder property, one can translate the properties of step skew products to the case of soft skew products.Theorem 2.
There exist diffeomorphismsf0 and f1 on any m-dimensional closed manifold M, and C2 neighborhoods U0(f0), U1(f1)⊂Diff2(M) such that, for each C>1 and α>0, if a soft skew product map G of the form (3) satisfies the following conditions:(1)
gω∈Uω0, for any ω∈Σ2,(2)
dC1(gω,gω′)≤C(dΣ2(ω,ω′))α, for ω,ω′∈Σ2,(3)
L·2-α<1,
then the periodic orbits ofG which are attracting (or repelling) along the fiber are dense in Σ2×M.Now by using the smooth realizations of step skew products, we prove that the above properties are preserved under small perturbations of these products in the space ofC2 diffeomorphisms.Theorem 3.
Letn and m be positive integers with n≥m+3, n≥5, and m≥1. Suppose that N is an n-dimensional closed manifold. Then there exists an open set 𝒰⊂Diff2(N) such that, for any f∈𝒰, there is a partially hyperbolic locally maximal invariant set Δ⊂N and two numbers l1 and l2=l1+m, such that the hyperbolic periodic orbits with stable manifolds of dimension li are dense in Δ.


## 2. Step Skew Products

This section is devoted to prove Theorem1. We will show that there exists an open set 𝒰 in the space of step skew product maps of the form (1) such that, for any map F∈𝒰, the periodic orbits of F which are attracting along M are dense in Σ2×M. The same property holds for periodic orbits which are repelling along M.First, let us recall some notations and definitions. We consider the iterations of step skew product mapF. Clearly, for n>0(7)Fn(ω,x)=(σnω,f¯n[w](x)),F-n(ω,x)=(σ-nω,f¯-n[w](x)),
where f¯n[ω]=fωn-1∘⋯∘fω0, f¯-n[ω]=fω-n-1∘⋯∘fω-1-1, f¯0[ω]=id. A periodic orbit of a step skew product map F is determined by its initial point (ω,x), where x∈M and ω∈Σ2 is a periodic sequence
(8)ω=⋯www⋯=(w),
with a finite zero-one segment w=(w0⋯wn-1). We say that a periodic orbit ((w),x) is attracting along  M if ∥Df¯|w|[w](x)∥<1 and is repelling along  M if ∥Df¯|w|[w](x)∥>1.From now on, the ambientM is a compact connected m-dimensional manifold without boundary. Also, let U,W⊂M be two disjoint open neighborhoods which are the domains of two local charts (W,φ), (U,ψ) of M. Take two gradient Morse-Smale vector fields on M, each of which possesses a unique hyperbolic repelling equilibrium qi and a unique hyperbolic attracting equilibrium pi, i=0,1, and finitely many saddle points rji, i=0,1, j=1,…,l, contained in open domains Vj⊂M∖(U∪W).Assume that the fixed pointsp0 and q1 are distinct points contained in U and p1 and q0 are also distinct points that are contained in W. Let f0 and f1 be their time-1 maps. Suppose that the mappings fi, i=0,1, have no saddle connection. Also, we can choose the coordinate functions φ and ψ satisfying the following conditions.(i)
If we takefi^:=ψ∘fi∘ψ-1, then fi^ are affine maps which are defined by
(9)f0^(x1,…,xm)=(±rxm+s,rx1,…,rxm-1),f1^(x1,…,xm)=(-ax1,ax2,…,axm-1,-axm-2sr),
for constants 0<r<1, 0<s<a-1, a>1 and ar<1. We consider a minus sign for even m and a plus sign for odd m. By construction,
(10)f0^∘f1^(x1,…,xm)=(±arxm-s,-arx1,arx2,…,arxm-1),
is a contracting map.(ii)
If we takefi~:=φ∘fi∘φ-1, i=0,1, then f0~=f0^-1 and f1~=f0^∘f1^-1∘f0^-1. So f0~-1=f0^ and (f0~∘f1~)-1=f0^∘f1^. Moreover, f1~ is an affine contracting map.Note that there is a compact invariant setΔ=Δℱ⊂U with nonempty interior which contains the fixed points p0 and q1, such that the acting of the iterated function system generated by {f0,f0∘f1} on Δ is minimal. Moreover, the iterated function system ℱ(M;f0,f1) is C1-robustly minimal (see [3] for more detail).Puth0:=f0 and h1:=f0∘f1. Let us define ℒ(Δ)=h0(Δ)∪h1(Δ). Suppose that Δin⊂Δ⊂Δout are two open sets close to Δ on which h0 and h1 are contracting. Then
(11)Δin⊂ℒ(Δin)⊂Δ⊂ℒ(Δout)⊂Δout,
and ℒi(Δin),ℒi(Δout) converge to Δ in the Hausdorff topology, as i→∞, provided that the fiber maps fi are sufficiently close to the identity map. This requires that the constants a and r are sufficiently close to 1.Moreover, our construction shows that the iterated function systemℱ(M;f0-1,f1-1) is also minimal. Also, there exists a compact invariant set Δ′=Δℱ′⊂W that contains the fixed points q0 and p1 in its interior such that the iterated function system ℱ(Δ′;f0-1,(f0∘f1)-1) is minimal. In particular, there exist open sets Δin′⊂Δ′⊂Δout′ satisfying the inclusion relations (11).In the rest of this section, we fix the mappingsfi, i=0,1, satisfying all the properties mentioned above and we consider the skew product map
(12)F:Σ2×M⟶Σ2×M,(ω,x)⟼(σω,fω0(x)),
with the fiber maps fi, i=0,1.In [3], the authors proved that F is C1-robustly topologically mixing on Σ112×Δ, where Σ112⊂Σ2 is the set of all sequences from Σ2 in which the segment “11” is not encountered to the right of any element.Sincefi, i=0,1, are Morse-Smale diffeomorphisms with a unique attracting fixed point pi and unique repelling fixed point qi and they have not any saddle connection, so the stable and unstable sets Ws(p0,f0) and Wu(q1,f1) are open and dense subsets of M.Lemma 4.
Consider the iterated function systemℱ(M;f0,f1) as aforementioned. For every nonempty open set U⊂M, there exist k≤k0∈ℕ and ρ=ρ(U)>0 such that, for every ball B⊂M of radius less than ρ, there exists a finite word w=t1⋯tk on the alphabets {0,1} and with the length k≤k0 such that f¯k[w](B)⊂U.Proof.
LetU⊂M be an open subset. Since the acting of ℱ on M is minimal, for each x∈M there exists a word w(x) on the alphabets {0,1} such that f¯|w(x)|[w(x)](x)∈U. By continuity, there is a neighborhood Vx of x such that f¯|w(x)|[w(x)](Vx)⊂U.
SinceM is compact, we can cover M by finitely many open sets Vxi, i=1,…,n. We take k0 as the maximum of the lengths of the words w(xi), i=1,…,n, and ρ>0 the Lebesgue number of this covering. Then every ball B⊂M of radius less than ρ is contained in some Vxi. So there exists a word w=t1⋯tk on the alphabets {0,1} of the length k≤k0 such that f¯k[w](B)⊂U.Remark 5.
Since the iterated function systemℱ(M;f0-1,f1-1) is minimal, we can apply the argument used in the proof of Lemma 4 to prove the following statement: for every nonempty open set U⊂M, there exists l≤l0∈ℕ and ϱ=ϱ(U)>0 such that, for every ball B⊂M of radius less than ϱ, there exists a finite word w=s1⋯sl on the alphabets {0,1} of the length l≤l0 such that fsl-1∘⋯∘fs1-1(B)⊂U.In the following, we will use the notation(13)Cα¯={ω∈Σ2∣ωj=αj,-n≤j≤n-1},
where α¯=α-n⋯α0⋯αn-1 is a segment of the symbols {0,1}.The rest of this section is devoted to prove Theorem1.Proof.
First, we will prove that the statement of Theorem1 holds for the step skew product map F with generators f0,f1 which are introduced in the aforementioned. Note that the open sets Cα¯×U⊂Σ2×M, form a base of the topology of the space Σ2×M where α¯=α-n⋯α0⋯αn-1 is a segment of {0,1}, Cα¯ is the cylinder set corresponding to the segment α¯, and U is an open set of M.
Suppose that the segmentα¯=α-n⋯α0⋯αn-1 and open subset U⊂M are given. We seek a periodic point ((β¯),x)∈Cα¯×U of the skew product map F which is attracting along M. From now on, we fix the open subset Cα¯×U⊂Σ2×M.
LetU0 be an open ball which is contained in the basin of the attracting fixed point p0 of f0 such that ∥Df0∣U0∥≤λ<1, for some 0<λ<1. By Lemma 4, there exist ρ0:=ρ0(U0) and k0:=k0(U0)∈ℕ such that, for every open neighborhood V of diameter less than ρ0, there exists a word w=w(V,U0) on the alphabets {0,1} and with the length at most k, such that f¯|w|[w](V)⊂U0.
Now the following statements hold.
(a) Consider an open ballW⊂U of radius less than ρ0/Ln. Take Wα+:=f¯n[α¯](W); then diam(Wα+)<ρ0. By Lemma 4, there exists a finite word w=t1⋯tl1 on the alphabets {0,1} of the length at most k0, such that f¯l1[w](Wα+)⊂U0.
(b) TakeWα-:=f¯-n[α¯](W). So there exist ρ2:=ρ2(Wα-) and k2:=k2(Wα-)∈ℕ satisfying the statement of Lemma 4.
SinceU0 is contained in the basin of attracting fixed point p0 of f0, so there exists a positive integer l2 such that
(14)diam(f0l2(f¯l1[w](Wα+)))<ρ2,Lk1+k2λl2<1.
By statement (b), there exists a word w′=s1⋯sl3 on the alphabets {0,1} and with the length l3≤k2 such that f¯l3[w′](f0l2(f¯l1[w](Wα+)))⊂Wα-.
We setβ¯=β-m⋯β-1β0⋯βm-1, where
(15)β-m⋯β-1=β0⋯βm-1=α0⋯αn-1t1…tl10⋯0︸l2timess1⋯sl3α-n⋯α-1
and m=l1+l2+l3+2n, which implies that f¯2m[β¯](W)⊂W. Moreover, the choice of l2 shows that ∥Df¯2m[β¯]∣W∥<1.
According to these facts, there exists an attracting fixed pointx for the mapping f¯2m[β¯] which is contained in W⊂U. So the periodic point ((β¯),x) which is attracting along the fiber lies in Cα¯×U.
Density of periodic orbits which are repelling alongM can be established similarly.
Indeed, by applying Remark5 and since the mapping f1-1 is contracting on Δ′, there exist an open set W⊂U and a finite word w′′=r1⋯rk on the alphabets {0,1}, such that
(16)frk-1∘⋯∘fr1-1∘f¯-n[α¯](W)⊂f¯n[α¯](W),∥(f¯n[α¯])-1∘frk-1∘⋯∘fr1-1∘f¯-n[α¯]∣W∥<1.
So there exists an attracting fixed point y for the map
(17)(fα-1∘⋯∘fα-n∘f¯k[w′′]∘f¯n[α¯])-1=(f¯n[α¯])-1∘frk-1∘⋯∘fr1-1∘f¯-n[α¯]
which is contained in W.
Now, we takeγ¯=γ-l⋯γ-1γ0⋯γl-1, where
(18)γ-l⋯γ-1=γ0⋯γl-1=α0⋯αn-1r1⋯rkα-n⋯α-1
and l=k+2n. Then ((γ¯),y) is a periodic point for the skew product map F which is repelling along M and lies in Cα¯×U.
Now, let us prove that the statement holds for small perturbations ofF, that is, step skew product maps generated by small perturbations of f0 and f1. Choose g0∈U0 and g1∈U1, sufficiently close to f0 and f1 and consider the step skew product map G given by (1) and with the fiber maps gi, i=0,1. Therefore, gi, i=0,1, possesses a unique hyperbolic repelling fixed point close to qi, i=0,1, a unique hyperbolic attracting fixed point close to pi, i=0,1, and finitely many saddle points which are close to rji, i=0,1, j=1,…,l. Moreover, the iterated function system 𝒢(M;g0,g1) is minimal and admits an invariant set Δ=Δ𝒢 with nonempty interior which contains the attracting fixed point of g0 and the repelling fixed of g1, such that 𝒢(Δ;g0,g0∘g1) is minimal. Moreover, the iterated function system 𝒢(M;g0-1,g1-1) is also minimal. So similar reasoning implies the existence of an attracting (repelling) periodic orbit for the map G which is contained in Cα¯×U. This terminates the proof of Theorem 1.


## 3. Soft Skew Products

In this section, we prove Theorem2. In fact, we describe the properties of soft skew product maps which have an m-dimensional closed manifold M as a fiber. To translate the properties of step skew product maps to the case of soft skew product maps, we need a Hölder property.In the following, we provide an open set in the space of soft systems (3) with the Hölder property that has the same properties of step systems.To be more precise, let us describe them in details.First, note that ifG is a soft skew product of the form (3), then it is obvious that, for n∈ℕ,
(19)Gn(ω,x)=(σnω,g¯n[w](x)),G-n(ω,x)=(σ-nω,g¯-n[w](x)),
where
(20)g¯n[ω]=gσn-1ω∘⋯∘gσω∘gω,g¯-n[ω]=gσ-nω-1∘⋯∘gσ-1ω-1,g¯0[ω]=id.
Let f0 and f1 be two diffeomorphisms on M generating a robustly minimal iterated function system as in the previous section. Write h0:=f0, h1:=f0∘f1 and let ℱ be the iterated function system generated by h0 and h1. Recall that the iterated function system ℱ acts minimally on a compact invariant set Δ. Also, there are open sets Δin⊂Δ⊂Δout on which
(21)Δin⊂ℱ(Δin)⊂Δ⊂ℱ(Δout)⊂Δout,
and h0 and h1 are contractions on Δout.Moreover, our construction in Section2 shows that the iterated function system ℱ(M;f0-1,f1-1) is also minimal. Also, there exists a compact invariant set Δ′ which contains the attracting fixed point of f1 and repelling fixed point of f0 in its interior such that the iterated function system ℱ(Δ′;f0-1,(f0∘f1)-1) is minimal. In particular, there exist open sets Δin′⊂Δ′⊂Δout′ satisfying the inclusion relations (21) corresponding to ℱ(Δ′;f0-1,(f0∘f1)-1).LetF on Σ2×M be defined by
(22)F(ω,x)=(σ(ω),hω0(x)),
where (σ(ω))k=ωk+1 is the left shift operator. Suppose that G is a soft skew product map of the form (3) such that gω depends continuously on ω and is uniformly close to hω0, by a uniform bound δ>0. Then the inclusions (21) get replaced by
(23)Σ2×Δin⊂G(Σ2×Δin),(Σ2×Δout)⊂Σ2×Δout,
for sufficiently small δ. Moreover, the choice of Δin can be independent of skew product map G. This means that if G is any soft skew product of the form (3) with the fiber maps gω, with dC1(gω,hω0)<δ, for any ω∈Σ2, then the inclusions (23) hold for G. By the argument used in the proof of [3, Proposition 5.1], the next lemma follows; see also [4, Proposition 5.1].Lemma 6.
LetF be the step skew product map as in the aforementioned and by fiber maps hi,i=0,1. Then any soft skew product map G of the form (3) which is sufficiently close to F possesses a maximal invariant set ΛG⊂Σ2×Δout on which the acting G is topologically mixing. Moreover, there is an open set Δin such that for any soft system G, Δin⊂π(ΛG), where π:Σ2×M→M is the natural projection.Since the diffeomorphismsfi, i=0,1, are Morse-Smale and the set of all Morse-Smale diffeomorphisms is open subset of Diff2(M), so we can choose two neighborhoods U0(f0), U1(f1)⊂Diff2(M) sufficiently small such that the following statements hold.IfG is a soft skew product of the form (3) with fiber maps gω∈Uω0(fω0), ω∈Σ2, then(i)
the mappinggω has one hyperbolic attracting fixed point p(ω), one hyperbolic repelling fixed point q(ω), and finitely many saddle points ri(ω), i=1,…,l;(ii)
all attracting fixed points of the mappingsgω, with ω0=0, and all repelling fixed points of the mappings gω, with ω0=1, lie strictly inside Δin;(iii)
all attracting fixed points of the mappingsgω, with ω0=1, and all repelling fixed points of the mappings gω, with ω0=0, lie strictly inside Δin′;(iv)
stable setsWs(pω,gω) are open and dense subsets of M, for any ω∈Σ2 with ω0=0;(v)
unstable setsWu(qω,gω) are open and dense subsets of M, for any ω∈Σ2 with ω0=1.We say that the soft skew product mapG is controllable if its fiber maps gω,ω∈Σ2, satisfying the assumptions of Theorem 2 and all of the properties mentioned above.In the following, we establish the density of periodic points of a controllable soft skew product mapG which are attracting along the fiber M.Indeed, we will find a periodic point in any open set of the formCα¯×U⊂Σ2×M, where α¯=α-n⋯α0⋯αn-1 is a finite segment of the alphabets {0,1}, Cα¯ is the cylinder set corresponding to it, and U is an open subset of M.First, we need the following lemma which controls the error in the coordinate along the fiber. It is obtained by an argument used in [1, Lemma  3.1].Lemma 7.
LetG be a controllable soft skew product map. Then there exists K>0, with K=K(L,C,α) and being independent of δ>0, such that, for any m∈ℕ, the inequality dΣ2(ω,ω′)≤2-m implies
(24)dC0(g¯±m[ω],g¯±m[ω′])≤γ:=Kδβ,
where β=1-lnL/ln2α.According to Lemma7, for each controllable soft skew product G with the fiber maps gω,
(25)diam{g¯±m[ω](x)∣ω={⋯w⋆⋯}}≤γ,
for any x∈M, any m∈ℕ, and any finite word w⋆=w-m⋯w-1·w0⋯wm.Let us note that ifδ>0 is sufficiently small, then γ>0 is also small enough. By Lemma 6, the controllable soft skew product G is topologically mixing on Σ112×Δ, where Σ112⊂Σ2 is the set of all sequences from Σ2 in which the segment “11” is not encountered to the right of any element.We now begin the proof of Theorem2.Proof.
Suppose that the segmentα¯=α-n⋯α0⋯αn-1 and open neighborhood U⊂M are given. Our aim is to find a periodic point in Cα¯×U, where Cα¯ is the cylinder set corresponding to α¯.
We recall that the stable setsWs(pω,gω) are open and dense subsets of manifold M, for any ω∈Σ2 with ω0=0, so
(26)g¯m[ω](U)∩Ws(pσmω,gσmω)≠∅,
for any m∈ℕ. This implies that there exists a neighborhood Uω1⊂U, such that g¯n[ω](Uω1)⊂Ws(pσnω,gσnω), for any sequence ω={⋯∣α0⋯αn-10⋯}.
Similarly,g¯n+1[ω](Uω1)∩Ws(pσn+1ω,gσn+1ω)≠∅, which implies that there is a neighborhood Uω2⊂Uω1, such that g¯n+1[ω](Uω2) is contained in Ws(pσn+1ω,gσn+1ω), for any sequence ω={⋯∣α0⋯αn-100⋯}.
By continuing the above procedure, we obtain neighborhoods(27)Uωk⊂Uωk-1⊂⋯Uω1⊂U
such that
(28)g¯n+k-1[ω](Uωk)⊂Ws(pσn+k-1ω,gσn+k-1ω),
for any sequence
(29)ω={⋯∣α0⋯αn-10⋯0︸k⋯}.
Since attracting fixed points of mappings gω, for any ω∈Σ2, are contained in Δin, so by increasing k, the subset g¯n+k[ω](Uωk) intersects with Δin. Therefore, there exists a positive integer k0 such that g¯n+k0[ω](Uωk0)∩Δin≠∅. Also, there is an open set U~ω⊂Uωk0 such that g¯n+k0[ω](U~ω)⊂Δin, for any sequence ω={⋯∣α0⋯αn-10⋯0︸k0⋯}.
By shrinkingg¯n+k0[ω](U~ω), we can control the error in the coordinate along the fiber. To do this, we note that the map gω, with ω0=0, and the map g¯2[ω], with ω0=1,ω1=0, are contracting on Δin, so there exists a finite word T=t1⋯tl1 such that g¯n+k0+l1[ω](U~ω) is contained in an open ball Uω+ of Δin with diameter 2γ, for any ω={⋯|α0⋯αn-10⋯0︸k0T⋯}.
Analogously, since the unstable subsetsWu(pω,gω) are open and dense subsets of manifold M, for any ω∈Σ2 with ω0=1, so
(30)g¯-m[ω](U)∩Ws(qσ-m-1ω,gσ-m-1ω-1)≠∅,
for any m∈ℕ and ω∈Σ2 with ω-m-1=1. This implies that there exists a neighborhood Wω1⊂U~ω, such that g¯-n[ω](Wω1)⊂Ws(qσ-n-1ω,gσ-n-1ω-1), for any sequence ω={⋯1α-n⋯α-1∣α0⋯αn-10⋯0︸k0T⋯}.
Similarly,g¯-n-1[ω](Wω1)∩Ws(qσ-n-2ω,gσ-n-2ω-1)≠∅, so there exists a neighborhood Wω2⊂Wω1, such that g¯-n-1[ω](Wω2)⊂Ws(qσ-n-2ω,gσ-n-2ω-1), for any ω={⋯11α-n⋯α-1∣α0⋯αn-10⋯0︸k0T⋯}.
By induction, we obtain neighborhoods(31)Wωm⊂Wωm-1⊂⋯Wω1⊂U~ω
such that
(32)g¯-n-m+1[ω](Wωm)⊂Ws(qσ-n-mω,gσ-n-mω-1),
for any sequence of the form
(33)ω={⋯1⋯1︸mα-n⋯α-1∣α0⋯αn-10⋯0︸k0T⋯}.
Since repelling fixed points of mappings gω, for any ω∈Σ2 with ω0=1, are contained in Δin, so by increasing m, g¯-n-m[ω](Wωm) intersects with Δin; therefore, there exist a positive integer m0 such that g¯-n-m0[ω](Wωm0)∩Δin≠∅ and an open set W~ω⊂Wωm0 such that g¯-n-m0[ω](W~ω)⊂Δin, for any sequence
(34)ω={⋯1⋯1︸m0α-n⋯α-1∣α0⋯αn-10⋯0︸k0T⋯}.
The construction shows that the mappingg¯-1[ω], with ω-1=0, and the mapping g¯-2[ω], with ω-1=0 and ω-2=1, are expanding on Δin⊂Δ, so there exists a finite word S=sl0⋯s1 such that, for any sequence ω of the form
(35)ω={⋯S1⋯1︸m0α-n⋯α-1∣α0⋯αn-10⋯0︸k0T⋯},g¯-(n+m0+l0)[ω](W~ω) contains an open ball Wω- of diameter 6γ.
Note that by shrinking theC2-neighborhoods U0(g0),U1(g1)⊂Diff2(M), if it is necessary, we may assume that 6γ<diam(Δin).
SinceW~ω⊂U~ω, the subset g¯n+k0+l1[ω](W~ω) is contained in an open ball Uω+ of Δin with diameter 2γ, for any sequence of the form
(36)ω={⋯S1⋯1︸m0α-n⋯α-1∣α0⋯αn-10⋯0︸k0T⋯}.
We recall that the acting ofG is topologically mixing on Σ112×Δ, so there exists a finite word Rω=r1⋯rkω∈Σ112,kω>k0, such that, for any sequence ω={⋯∣Rω⋯},Bγ(g¯kω[ω](Uω+))⊂Wω-.
Take the segment(37)w:=S1⋯1︸m0α-n⋯α-1∣α0⋯αn-10⋯0︸k0TR
and the periodic sequence β~:=(w).
Now the constructions show thatg¯n+k0+l1[β~](W~β~) is contained in an open ball Uβ~+ in Δin of diam 2γ, and g¯-(n+m0+l0)[β~](W~β~) contains an open ball Wβ~- of diam 6γ. So
(38)(g¯-(n+m0+l0)[β~])-1(Wβ~-)=g¯n+m0+l0[σ-(n+m0+l0)β~](Wβ~-)⊂W~β~.
Letm=2n+l0+l1+m0+K0+k. According to Lemma 7 and the fact g¯kβ~[σn+k+l1β~](Uβ~+)⊂Wβ~-, we conclude that
(39)g¯m[β~](W~β~)⊂W~β~.
Note that the acting ofgω, with ω0=0, and gω′, with ω0′=1 and ω1′=0, are contracting on Δin, so we can choose kβ~ sufficiently large such that ∥Dg¯m[β~]∥<1 on W~β~.
Hence,g¯m[β~] has an attracting fixed point y~∈W~β~. So Y~=((β~),y~) is a periodic point in Cα¯×U which is attracting along the fiber. By a similar argument, we conclude the existence of a periodic point in Cα¯×U which is repelling along the fiber. This completes the proof of Theorem 2.


## 4. Perturbations

Letn and m be positive integers with n≥m+3, n≥5, and m≥1. Suppose that N is an n-dimensional closed manifold. In this section, we will construct an open set 𝒰 of Diff2(N) that satisfies the following property: each diffeomorphism of 𝒰 possesses a partially hyperbolic locally maximal invariant set with a dense subset of periodic points with different indices.In fact, we will find diffeomorphisms such that the restriction of them to their locally maximal invariant sets is conjugated to step random dynamical systems of the form (1).As we have mentioned before, many properties observed for these products appear to persist as properties of diffeomorphisms [1, 2].In the following, first we need to introduce skew products over the horseshoe which can be considered as smooth realizations of skew products over the Bernoulli shift of the forms (1) and (3).Indeed, suppose thath:S2→S2 is a diffeomorphism with a horseshoe type hyperbolic set Λ, which has a Markov partition with two rectangles D0, D1 such that D0∩D1=∅, with the rate of contraction k∈(0,1) which is small enough (see [1, Theorem  2]). Put D:=D0∪D1 and h(D):=D′. It is well known that the hyperbolic invariant set Λ is homeomorphic to Σ2 with restriction of h to Λ being conjugate to the Bernoulli shift σ on Σ2.Now we define a skew product over the horseshoe maph:Λ→Λ with the fiber map M as follows:
(40)ℱ:D×M⟶D′×M,ℱ∣Di×M=h×fi,i=0,1,
where the diffeomorphism fi:M→M, i=0,1, are the generators of a skew products F of the form (1). The skew product ℱ is called a smooth realization of the skew product F. It is easy to see that Λ×M is partially hyperbolic for ℱ and ℱ∣Λ×M is conjugate to step skew product F. This fact implies that the properties found during the investigation of a semigroup generated by the diffeomorphisms fi:M→M are realized by smooth mapping ℱ.Suppose that𝒢 is a C2 skew product which is C1-close to ℱ. Then 𝒢 has an invariant set 𝒴𝒢 homeomorphic to Σ2×M by a homeomorphism K (see [2]). Let π:Σ2×M→M be the projection to the fiber along the base. The homeomorphism K:Σ2×M→𝒴𝒢, 𝒴𝒢⊂D×M, can be taken so that the coordinate x is preserved, and hence the restriction of K to a single fiber is a C2-diffeomorphism. One can consider the induced mapping
(41)G=K-1∘𝒢∘K:Σ2×M⟶Σ2×M.
Let us denote the mapping π∘K-1∘𝒢∘K(ω,·):M→M by gω which depends on ω. Then gω is C2 and the mapping G has the following form:
(42)G:Σ2×M⟶Σ2×M,(ω,x)⟶(σω,gω(x)),
which is a soft skew product (see [2] for more detail). We say that G is a soft skew product corresponding to 𝒢 or 𝒢 is a k-realization of G. Moreover, the bundle map gω is C1-close to fω0 for each ω∈Σ2.Here, we takeM=Sm, the m-dimensional sphere. Let f0 and f1 be two diffeomorphisms on Sm generating a robustly minimal iterated function system as in Sections 2 and 3. Also, let F be the step skew product map of the form (1) with the fiber maps f0 and f1, and let ℱ be its smooth realization. Let us take neighborhoods U0,U1 as in Theorem 1.Now, let𝒢 be C1-close to ℱ. Then 𝒢 is conjugate to a controllable soft skew product map G, with fiber maps gω which is C1-close to fω0; see Section 3 for more detail.Letℋ be a C2 diffeomorphism which is C1-close to 𝒢. Then, ℋ has an invariant set 𝒴ℋ homeomorphic to Σ2×Sm such that the projection (𝒴ℋ,ℋ)↦(Σ2,σ) is semiconjugacy and so the dynamics of ℋ restricted to 𝒴𝒢 resembles the dynamics of ℱ∣Λ×Sm. Also, ℋ restricted to 𝒴ℋ is conjugate to skew product H on Σ2×Sm (see [2]). In particular, the fiber maps hω are C1-close to gω and therefore it is C1-close to fω0, for each ω∈Σ2.Now, we can apply Theorem2 to conclude that the periodic orbits of the skew product H which are attracting (repelling) along Sm are dense in Σ2×Sm. Therefore, ℋ restricted to 𝒴ℋ has a dense subset of periodic orbits of indices (dimension of their stable manifolds) l1=1 and l2=m+1.Finally, one can see thatℋ restricted to 𝒴ℋ can be extended to a diffeomorphism on the closed manifold N.Indeed, one can embed them-sphere Sm in ℝm+1 and a two-dimensional rectangle B in ℝn-m-1, where D⊂B, D=D0∪D1. So B×Sm can be embedded in the closed manifold N, by a local chart of N (see [2] for more detail). This completes the proof of Theorem 3.



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*Source: 539736-2013-10-03.xml*