#### Abstract

We consider step and soft skew products over the Bernoulli shift which have an -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.

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

Consider diffeomorphisms , , defined on . The *iterated function system* is the semigroup generated by , that is, the set of all maps , where .

The -*orbit* of is the set of points , .

An iterated function system is called *minimal* if each closed subset with , for all , is empty or coincides with . This means that -orbit of each is dense in .

Let , , be diffeomorphisms of . A *step skew product* over the Bernoulli shift is defined by
where is the space of two-sided sequences of 2 symbols . Consider the following standard metric on :
where and .

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

A *soft skew product* over the Bernoulli shift is a map
where the fiber maps 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 sequence .

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 be a finite segment on the alphabets . We denote by an arbitrary infinite sequence in which occurs starting from the zeroth position. In a similar way, we introduce the notation and . We also denote by the length of .

We recall that a map is called *topologically mixing* if for each nonempty open sets , intersects with for all large enough .

For a diffeomorphism of , a compact -invariant set has a *dominated splitting* if
where each is nontrivial and -invariant for and there exists an such that
for every , and .

The set is *partially hyperbolic* if it has a dominated splitting
and there exists some such that either uniformly contracts or uniformly expands .

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 exist diffeomorphisms , , and -neighborhoods , such that for any and , the periodic orbits of the step skew product of the form (1) with the fiber maps , which are attracting (or repelling) along , are dense in .*

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 diffeomorphisms and on any -dimensional closed manifold , and neighborhoods , such that, for each and , if a soft skew product map of the form (3) satisfies the following conditions:*(1)*, for any ,*(2)*, for ,*(3)*,**
then the periodic orbits of which are attracting (or repelling) along the fiber are dense in .*

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 diffeomorphisms.

Theorem 3. *Let and be positive integers with , , and . Suppose that is an -dimensional closed manifold. Then there exists an open set such that, for any , there is a partially hyperbolic locally maximal invariant set and two numbers and , such that the hyperbolic periodic orbits with stable manifolds of dimension are dense in .*

#### 2. Step Skew Products

This section is devoted to prove Theorem 1. We will show that there exists an open set in the space of step skew product maps of the form (1) such that, for any map , the periodic orbits of which are attracting along are dense in . The same property holds for periodic orbits which are repelling along .

First, let us recall some notations and definitions. We consider the iterations of step skew product map . Clearly, for
where , , . A periodic orbit of a step skew product map is determined by its initial point , where and is a periodic sequence
with a finite zero-one segment . We say that a periodic orbit is *attracting along* if and is *repelling along* if .

From now on, the ambient is a compact connected -dimensional manifold without boundary. Also, let be two disjoint open neighborhoods which are the domains of two local charts , of . Take two gradient Morse-Smale vector fields on , each of which possesses a unique hyperbolic repelling equilibrium and a unique hyperbolic attracting equilibrium , , and finitely many saddle points , , , contained in open domains .

Assume that the fixed points and are distinct points contained in and and are also distinct points that are contained in . Let and be their time-1 maps. Suppose that the mappings , , have no saddle connection. Also, we can choose the coordinate functions and satisfying the following conditions.(i)If we take , then are affine maps which are defined by for constants , , and . We consider a minus sign for even and a plus sign for odd . By construction, is a contracting map.(ii)If we take , , then and . So and . Moreover, is an affine contracting map.

Note that there is a compact invariant set with nonempty interior which contains the fixed points and , such that the acting of the iterated function system generated by on is minimal. Moreover, the iterated function system is -robustly minimal (see [3] for more detail).

Put and . Let us define . Suppose that are two open sets close to on which and are contracting. Then and converge to in the Hausdorff topology, as , provided that the fiber maps are sufficiently close to the identity map. This requires that the constants and are sufficiently close to 1.

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

In the rest of this section, we fix the mappings , , satisfying all the properties mentioned above and we consider the skew product map with the fiber maps , .

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

Since , , are Morse-Smale diffeomorphisms with a unique attracting fixed point and unique repelling fixed point and they have not any saddle connection, so the stable and unstable sets and are open and dense subsets of .

Lemma 4. *Consider the iterated function system as aforementioned. For every nonempty open set , there exist and such that, for every ball of radius less than , there exists a finite word on the alphabets and with the length such that .*

*Proof. *Let be an open subset. Since the acting of on is minimal, for each there exists a word on the alphabets such that . By continuity, there is a neighborhood of such that .

Since is compact, we can cover by finitely many open sets , . We take as the maximum of the lengths of the words , , and the Lebesgue number of this covering. Then every ball of radius less than is contained in some . So there exists a word on the alphabets of the length such that .

*Remark 5. *Since the iterated function system is minimal, we can apply the argument used in the proof of Lemma 4 to prove the following statement: for every nonempty open set , there exists and such that, for every ball of radius less than , there exists a finite word on the alphabets of the length such that .

In the following, we will use the notation where is a segment of the symbols .

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

*Proof. *First, we will prove that the statement of Theorem 1 holds for the step skew product map with generators which are introduced in the aforementioned. Note that the open sets , form a base of the topology of the space where is a segment of , is the cylinder set corresponding to the segment , and is an open set of .

Suppose that the segment and open subset are given. We seek a periodic point of the skew product map which is attracting along . From now on, we fix the open subset .

Let be an open ball which is contained in the basin of the attracting fixed point of such that , for some . By Lemma 4, there exist and such that, for every open neighborhood of diameter less than , there exists a word on the alphabets and with the length at most , such that .

Now the following statements hold.

(a) Consider an open ball of radius less than . Take ; then . By Lemma 4, there exists a finite word on the alphabets of the length at most , such that .

(b) Take . So there exist and satisfying the statement of Lemma 4.

Since is contained in the basin of attracting fixed point of , so there exists a positive integer such that
By statement , there exists a word on the alphabets and with the length such that .

We set , where
and , which implies that . Moreover, the choice of shows that .

According to these facts, there exists an attracting fixed point for the mapping which is contained in . So the periodic point which is attracting along the fiber lies in .

Density of periodic orbits which are repelling along can be established similarly.

Indeed, by applying Remark 5 and since the mapping is contracting on , there exist an open set and a finite word on the alphabets , such that
So there exists an attracting fixed point for the map
which is contained in .

Now, we take , where
and . Then is a periodic point for the skew product map which is repelling along and lies in .

Now, let us prove that the statement holds for small perturbations of , that is, step skew product maps generated by small perturbations of and . Choose and , sufficiently close to and and consider the step skew product map given by (1) and with the fiber maps , . Therefore, , , possesses a unique hyperbolic repelling fixed point close to , , a unique hyperbolic attracting fixed point close to , , and finitely many saddle points which are close to , , . Moreover, the iterated function system is minimal and admits an invariant set with nonempty interior which contains the attracting fixed point of and the repelling fixed of , such that is minimal. Moreover, the iterated function system is also minimal. So similar reasoning implies the existence of an attracting (repelling) periodic orbit for the map which is contained in . This terminates the proof of 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 -dimensional closed manifold 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 if is a soft skew product of the form (3), then it is obvious that, for , where Let and be two diffeomorphisms on generating a robustly minimal iterated function system as in the previous section. Write , and let be the iterated function system generated by and . Recall that the iterated function system acts minimally on a compact invariant set . Also, there are open sets on which and and are contractions on .

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

Let on be defined by where is the left shift operator. Suppose that is a soft skew product map of the form (3) such that depends continuously on and is uniformly close to , by a uniform bound . Then the inclusions (21) get replaced by for sufficiently small . Moreover, the choice of can be independent of skew product map . This means that if is any soft skew product of the form (3) with the fiber maps , with , for any , then the inclusions (23) hold for . By the argument used in the proof of [3, Proposition 5.1], the next lemma follows; see also [4, Proposition 5.1].

Lemma 6. *Let be the step skew product map as in the aforementioned and by fiber maps . Then any soft skew product map of the form (3) which is sufficiently close to possesses a maximal invariant set on which the acting is topologically mixing. Moreover, there is an open set such that for any soft system , , where is the natural projection. *

Since the diffeomorphisms , , are Morse-Smale and the set of all Morse-Smale diffeomorphisms is open subset of , so we can choose two neighborhoods , sufficiently small such that the following statements hold.

If is a soft skew product of the form (3) with fiber maps , , then(i)the mapping has one hyperbolic attracting fixed point , one hyperbolic repelling fixed point , and finitely many saddle points , ;(ii)all attracting fixed points of the mappings , with , and all repelling fixed points of the mappings , with , lie strictly inside ;(iii)all attracting fixed points of the mappings , with , and all repelling fixed points of the mappings , with , lie strictly inside ;(iv)stable sets are open and dense subsets of , for any with ;(v)unstable sets are open and dense subsets of , for any with .

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

Indeed, we will find a periodic point in any open set of the form , where is a finite segment of the alphabets , is the cylinder set corresponding to it, and is an open subset of .

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. *Let be a controllable soft skew product map. Then there exists , with and being independent of , such that, for any , the inequality implies
**
where . *

According to Lemma 7, for each controllable soft skew product with the fiber maps , for any , any , and any finite word .

Let us note that if is sufficiently small, then is also small enough. By Lemma 6, the controllable soft skew product is topologically mixing on , where is the set of all sequences from in which the segment “11” is not encountered to the right of any element.

We now begin the proof of Theorem 2.

*Proof. *Suppose that the segment and open neighborhood are given. Our aim is to find a periodic point in , where is the cylinder set corresponding to .

We recall that the stable sets are open and dense subsets of manifold , for any with , so
for any . This implies that there exists a neighborhood , such that , for any sequence .

Similarly, , which implies that there is a neighborhood , such that is contained in , for any sequence .

By continuing the above procedure, we obtain neighborhoods
such that
for any sequence
Since attracting fixed points of mappings , for any , are contained in , so by increasing , the subset intersects with . Therefore, there exists a positive integer such that . Also, there is an open set such that , for any sequence .

By shrinking , we can control the error in the coordinate along the fiber. To do this, we note that the map , with , and the map , with , are contracting on , so there exists a finite word such that is contained in an open ball of with diameter , for any .

Analogously, since the unstable subsets are open and dense subsets of manifold , for any with , so
for any and with . This implies that there exists a neighborhood , such that , for any sequence .

Similarly, , so there exists a neighborhood , such that , for any .

By induction, we obtain neighborhoods
such that
for any sequence of the form
Since repelling fixed points of mappings , for any with , are contained in , so by increasing , intersects with ; therefore, there exist a positive integer such that and an open set such that , for any sequence

The construction shows that the mapping , with , and the mapping , with and , are expanding on , so there exists a finite word such that, for any sequence of the form
contains an open ball of diameter .

Note that by shrinking the -neighborhoods , if it is necessary, we may assume that .

Since , the subset is contained in an open ball of with diameter , for any sequence of the form

We recall that the acting of is topologically mixing on , so there exists a finite word , such that, for any sequence .

Take the segment
and the periodic sequence .

Now the constructions show that is contained in an open ball in of diam , and contains an open ball of diam . So

Let . According to Lemma 7 and the fact , we conclude that

Note that the acting of , with , and , with and , are contracting on , so we can choose sufficiently large such that on .

Hence, has an attracting fixed point . So is a periodic point in which is attracting along the fiber. By a similar argument, we conclude the existence of a periodic point in which is repelling along the fiber. This completes the proof of Theorem 2.

#### 4. Perturbations

Let and be positive integers with , , and . Suppose that is an -dimensional closed manifold. In this section, we will construct an open set of that satisfies the following property: each diffeomorphism of 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 is a diffeomorphism with a horseshoe type hyperbolic set , which has a Markov partition with two rectangles , such that , with the rate of contraction which is small enough (see [1, Theorem 2]). Put and . It is well known that the hyperbolic invariant set is homeomorphic to with restriction of to being conjugate to the Bernoulli shift on .

Now we define a skew product over the horseshoe map with the fiber map as follows:
where the diffeomorphism , , are the generators of a skew products of the form (1). The skew product is called a *smooth realization* of the skew product . It is easy to see that is partially hyperbolic for and is conjugate to step skew product . This fact implies that the properties found during the investigation of a semigroup generated by the diffeomorphisms are realized by smooth mapping .

Suppose that is a skew product which is -close to . Then has an invariant set homeomorphic to by a homeomorphism (see [2]). Let be the projection to the fiber along the base. The homeomorphism , , can be taken so that the coordinate is preserved, and hence the restriction of to a single fiber is a -diffeomorphism. One can consider the induced mapping Let us denote the mapping by which depends on . Then is and the mapping has the following form: which is a soft skew product (see [2] for more detail). We say that is a soft skew product corresponding to or is a -realization of . Moreover, the bundle map is -close to for each .

Here, we take , the -dimensional sphere. Let and be two diffeomorphisms on generating a robustly minimal iterated function system as in Sections 2 and 3. Also, let be the step skew product map of the form (1) with the fiber maps and , and let be its smooth realization. Let us take neighborhoods as in Theorem 1.

Now, let be -close to . Then is conjugate to a controllable soft skew product map , with fiber maps which is -close to ; see Section 3 for more detail.

Let be a diffeomorphism which is -close to . Then, has an invariant set homeomorphic to such that the projection is semiconjugacy and so the dynamics of restricted to resembles the dynamics of . Also, restricted to is conjugate to skew product on (see [2]). In particular, the fiber maps are -close to and therefore it is -close to , for each .

Now, we can apply Theorem 2 to conclude that the periodic orbits of the skew product which are attracting (repelling) along are dense in . Therefore, restricted to has a dense subset of periodic orbits of indices (dimension of their stable manifolds) and .

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

Indeed, one can embed the -sphere in and a two-dimensional rectangle in , where , . So can be embedded in the closed manifold , by a local chart of (see [2] for more detail). This completes the proof of Theorem 3.

#### Acknowledgment

The authors are very grateful to the referee for fruitful comments and valuable suggestions.