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

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 an m-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 space M will be an m-dimensional closed manifold and its metric is geodesic distance and the measure is the Riemannian volume.

Consider diffeomorphisms f_i, i = 1, …, k, defined on M. The iterated function system  ℱ(M; f₁, …, f_k) is the semigroup generated by f₁, …, f_k, that is, the set of all maps $$, where t_j, …, t₁ ∈ {1, …, k}.

The ℱ-orbit of x ∈ M is the set of points $$, t_j ≥ 0.

An iterated function system F(M; f₁, …, f_k) is called minimal if each closed subset A with f_i(A) ⊂ A, for all i, is empty or coincides with M. This means that ℱ-orbit of each x ∈ M is dense in M.

Let us note that an iterated function system can be embedded in a single dynamical system, the skew product F of the form (1), such that the action orbits of the iterated function system ℱ with generators f_i coincide with the projections of positive semitrajectories of the skew product F onto the fiber along the base.

We would like to mention that in contrast to step skew products, the fiber maps of soft skew products depend on the whole sequence ω.

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].

Let w 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 map F 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.

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.

By applying the Hölder property, one can translate the properties of step skew products to the case of soft skew products.

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 of C² diffeomorphisms.

2. Step Skew Products

This section is devoted to prove Theorem 1. 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.

From now on, the ambient M 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 q_i and a unique hyperbolic attracting equilibrium p_i, i = 0,1, and finitely many saddle points $$, i = 0,1, j = 1, …, l, contained in open domains V_j ⊂ M∖(U ∪ W).

Note that there is a compact invariant set Δ = Δ_ℱ ⊂ 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).

Moreover, our construction shows that the iterated function system $$ is also minimal. Also, there exists a compact invariant set $$ that contains the fixed points q₀ and p₁ in its interior such that the iterated function system $$ is minimal. In particular, there exist open sets $$ satisfying the inclusion relations (11).

In [3], the authors proved that F is C¹-robustly topologically mixing on $$, where $$ is the set of all sequences from Σ² in which the segment “11” is not encountered to the right of any element.

Since f_i, i = 0,1, are Morse-Smale diffeomorphisms with a unique attracting fixed point p_i and unique repelling fixed point q_i and they have not any saddle connection, so the stable and unstable sets W^s(p₀, f₀) and W^u(q₁, f₁) are open and dense subsets of M.

The rest of this section is devoted to prove Theorem 1.

3. Soft Skew Products

In this section, we prove Theorem 2. 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.

Moreover, our construction in Section 2 shows that the iterated function system $$ is also minimal. Also, there exists a compact invariant set Δ^′ which contains the attracting fixed point of f₁ and repelling fixed point of f₀ in its interior such that the iterated function system $$ is minimal. In particular, there exist open sets $$ satisfying the inclusion relations (21) corresponding to $$.

Since the diffeomorphisms f_i, 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.

We say that the soft skew product map G is controllable if its fiber maps g_ω,   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 map G which are attracting along the fiber M.

Indeed, we will find a periodic point in any open set of the form $$, where $$ is a finite segment of the alphabets {0,1}, $$ 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].

Let us note that if δ > 0 is sufficiently small, then γ > 0 is also small enough. By Lemma 6, the controllable soft skew product G is topologically mixing on $$, where $$ 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 Theorem 2.

4. Perturbations

Let n 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 that h : 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.

Here, we take M = S^m, the m-dimensional sphere. Let f₀ and f₁ be two diffeomorphisms on S^m 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, let 𝒢 be C¹-close to ℱ. Then 𝒢 is conjugate to a controllable soft skew product map G, with fiber maps g_ω which is C¹-close to $$; see Section 3 for more detail.

Let ℋ be a C² diffeomorphism which is C¹-close to 𝒢. Then, ℋ has an invariant set 𝒴_ℋ homeomorphic to Σ² × S^m such that the projection (𝒴_ℋ, ℋ)↦(Σ², σ) is semiconjugacy and so the dynamics of ℋ restricted to 𝒴_𝒢 resembles the dynamics of $$. Also, ℋ restricted to 𝒴_ℋ is conjugate to skew product H on Σ² × S^m (see [2]). In particular, the fiber maps h_ω are C¹-close to g_ω and therefore it is C¹-close to $$, for each ω ∈ Σ².

Now, we can apply Theorem 2 to conclude that the periodic orbits of the skew product H which are attracting (repelling) along S^m are dense in Σ² × S^m. Therefore, ℋ restricted to 𝒴_ℋ has a dense subset of periodic orbits of indices (dimension of their stable manifolds) l₁ = 1 and l₂ = m + 1.

Finally, one can see that ℋ restricted to 𝒴_ℋ can be extended to a diffeomorphism on the closed manifold N.

Indeed, one can embed the m-sphere S^m in ℝ^m+1 and a two-dimensional rectangle B in ℝ^n−m−1, where D ⊂ B, D = D₀ ∪ D₁. So B  ×  S^m 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.