Inverse Limit Spaces with Various Shadowing Property

Abstract

We discuss the relationship between ergodic shadowing property and inverse shadowing property of and that of the shift map σ_f on the inverse limit space.

1. Introduction

The notion of pseudoorbits and pseudoorbit shadowing property often appears in several branches of modern theory of dynamical systems (see [1]) and usually plays an important role in the investigation of stability theory and ergodic theory. The authors [2] introduced the notion of ergodic shadowing for a continuous onto map which is equivalent to the map being topologically mixing and has the ordinary shadowing property and deduces the chaotic behavior of a map with ergodic shadowing property. In [3], the authors proved that if any -ergodic pseudoorbit of a system is shadowed by a point along a set of positive lower density, then this system is chain mixing and if it is minimal, then it is topologically weakly mixing and so is Li-Yorke chaos. Recently, in [4], the notion of weakly ergodic shadowing property was introduced; the author showed that weakly ergodic shadowing property is equivalent to ergodic shadowing property. If has ergodic shadowing property (ESP), then it has shadowing property (SP), but as we will see the converse is not true; that is, there are systems with shadowing property which do not have ergodic shadowing property (see Example 3). Therefor ergodic shadowing property is not equivalent to shadowing property. Inverse limits of maps have been studied for decades, as a means of studying both continua and dynamical systems. In 2004 and 2006, Mahavier [5] and Ingram and Mahavier [6] introduced the concept of inverse limits with upper semicontinuous set-valued functions. Inverse limit is a useful tool to study the dynamical properties of smooth systems; some dynamical properties of can be interpreted by the topological structures of the inverse limit dynamical system; for example, Chen and Li [7] proved that has the shadowing property if and only if has so. Li [8] proved that some dynamical properties hold simultaneously for both and . In this paper, we discuss the relationship between ergodic shadowing property and inverse shadowing property for a surjective continuous map on a compact metric space and shift map on the inverse limit space.

2. Definitions

Let be a compact metric space with metric and let be a continuous map. For every , is defined inductively by , where is the identity map on . For , a sequence is called a -pseudoorbit of if for every . If, for any and every , there is a finite -pseudoorbit of such that , , then is called an -chain from to , and is called the length of the -chain. If, for any and every , there is an -chain from to , then is called chain transitive. Moreover if there is a positive integer , such that when , there is an -chain with length from to , then is called chain mixing. For any nonempty open sets and if there is such that , then is called topologically transitive and if there is , such that, for any , , then is called topologically mixing. Obviously, topologically transitive (mixing) map is chain transitive (mixing). Let be given. We write if for every there is a -pseudoorbit of some length of such that . We write if and . A sequence (resp., ) in is said to be -traced by some point in if for each integer (resp., for every ). A map is said to have the shadowing property, SP (resp., ), if for any there is such that every -pseudoorbit of can be -shadowed by some point in . Let be the compact metric space of all two-sided sequences in , endowed with the product topology. For a constant and , let denote the set of all -pseudoorbits of . A mapping satisfying , , is said to be a -method for . For convenience, write for .

is said to be inverse shadowing, denoted by ISP (resp., positive inverse shadowing property, ) if for any there exists such that for any and any -method (resp., ) there is such that , for all (resp., for all ).

Given a sequence (resp., ), put (resp., ) and  (resp., ).

For a sequence (resp., ) and a point of , put (resp., ) and (resp., (resp., ) is called a -ergodic pseudoorbit for if (resp., . A -ergodic pseudoorbit is said to be -ergodic shadowed by a point in , denoted by ESP (resp., ) if (resp., ). For an-invariant set , we say that has ergodic shadowing property (or is ergodic shadowable for ) if for every there exists such that every -ergodic pseudoorbit in could be -ergodic shadowed by a point in . The following lemma is proved in [2].

Proposition 1. Suppose be a continuous mapping and surjective on compact metric space . For the dynamical system (), the following properties are equivalent: (a) ergodic shadowable and (b) shadowable and topological mixing.

We do not recall here the definition of topological entropy, since it is well known (e.g., see [9]). For our purpose it is only important to remember that topological entropy is a number and for every transitive map it is at least (see Corollary 3.6 in [10]). It is known that every one-dimensional compact connected manifold is homeomorphic with or . The following proposition shows that there exists no homeomorphism of or with ESP. thus in this paper the space cannot be a one-dimensional connected compact space.

Proposition 2. (1) There exists no homeomorphism with topological mixing.
(2) Every homeomorphism has entropy 0.

Proof. We prove item but (2) follows from Theorem 7.14 in [11]. Let be a continuous surjective map which is topologically mixing. Since is onto, there exists such that . Let . Choose open intervals such that, for every and , . For open intervals there exists such that if , then and . Thus there exist and such that and . Hence using intermediate value theorem there exists such that ; therefore is not one to one. If or , since is topologically mixing, it is easy to see that there exists another fixed point in ; thus, in any case, is not one to one.

We know that if has ergodic shadowing property, then it has one chain component and SP. But the converse is not true. To see this we use an example of [12] to introduce a map which is shadowable but does not have ergodic shadowing property.

Example 3. Let , with the following metric: Let be a permutation of the set for some . Let if , and otherwise. is a homeomorphism and every point of is a periodic point for . Let be arbitrary and consider . Then if and only if , where . Hence if and only if . So the chain components of are the singletons, and hence does not have ergodic shadowing property. But one can easily see that has the shadowing property.

Proposition 4. If is continuous and has only one fixed point at endpoint interval, then has .

To prove the proposition, we need the following lemma; see ([7]).

Lemma 5. Let be a compact metric space and let be a continuous map. Given and , there exists such that every -pseudoorbit satisfies for all .

Proof of Proposition 4. Let and for every , . For every , let and for ; thus and for all . If , then . Let ; we have for every and . By Lemma 5 for and there exists such that if is a -pseudoorbit of , then ; thus . By the uniform continuity of , there exists such that every -pseudoorbit satisfies for every and there exists such that if , then for . Choose ; let be a -method of . If , then for , we have . Since for , such that . Since for , such that . But for and -pseudoorbit , ; this means that, for , .

The following proposition is proved in [7].

Proposition 6. If is continuous and has fixed points only at the end of interval, then has shadowing property.

It is known from [13] that the shift homeomorphism has shadowing property if and only if the sole bonding map does. Recently in [14], the author discussed connection between limit shadowing property for a continuous map on a compact metric space and that for the shift map on the inverse limit space. In next section we argue relation between ergodic shadowing property (inverse shadowing property) and .

3. Results

Let be a compact metric space. Consider the compact metric space of all two-sided sequences , endowed with the product topology. Let be a continuous map on . Then the closed subspace , for all of together with the associated shift map defined by with for all is called the inverse limit space of . Note that, with the metric defined by , is a compact metric space. For every define the project map by for every such that . is an open continuous map and satisfies for every .

Theorem 7. Let be a continuous surjective map on a compact metric space . (1)If has ergodic shadowing property, then the shift map on the inverse limit space has ergodic shadowing property.(2)If has inverse shadowing property, then the shift map on the inverse limit space has inverse shadowing property.

Proof. Given , suppose that diam and choose such that . Uniform continuity of implies existence of some such that if , then for .
Since has ergodic shadowing property, for , there exists such that every -ergodic pseudoorbit is -ergodic shadowed by some points of . Choose such that . Let be -ergodic pseudoorbit for . Since the sequence is a -ergodic pseudoorbit for . But there exists such that
Let for and for ; then . It is easy to check that if , then ; that is, . Therefor (2) shows that can be -ergodic shadowed by .
Since has inverse shadowing property, for , there exists such that, for any and any -method , there is such that , for all . Choose such that . Suppose that is a -method for ; we construct a -method for as follows: let ; choose a point such that . Since is a -method for , therefor, for , . Consider such that , where . It is easy to see that is a -pseudoorbit for ; thus is a -method for . Since has inverse shadowing property, for , such that , there exists such that . Let such that ; then ; this means that has inverse shadowing property.

The converse of the above theorem is not true. We use an example of [13] to show that in general the ergodic shadowing of on does not imply the ergodic shadowing of on .

Example 8. Consider with , , , , , , and such that is linear on each of the subintervals , , , , , , and . Observe that is not shadowing as has no shadowing property; thus has no ergodic shadowing property. Further, a point if and only if for all . Therefor shift map has only two fixed points; thus, by Proposition 6, has shadowing property and it is known that shift map is topologically mixing (see proposition 7.5 of [9]); therefor has ergodic shadowing property. Note that is not a local homeomorphism on .
We do not know whether the converse of Theorem 7 in the case of inverse shadowing property holds. But we give an example that has but has no .

Example 9. Consider with , , , , and , such that is linear on each of the subintervals , , , , and . Observe that has no , but has only one fixed point; thus, by Proposition 4, it is positive inverse shadowing, . Note that is not a local homeomorphism.
In the following theorem we express some conditions under which ergodic shadowing property and inverse shadowing property of on imply ergodic shadowing property and inverse shadowing property of on , respectively.

Theorem 10. Let be a local homeomorphism on a compact metric space . (1)If the shift map on has ergodic shadowing property, then has ergodic shadowing property.(2)If the shift map on has inverse shadowing property, then has inverse shadowing property.

Proof. Given , by ergodic shadowing property of , there exists such that every -ergodic pseudoorbit for is -ergodic shadowed by some point in . Let = diam; choose such that . is a local homeomorphism; there exists with such that is a homeomorphism, where is the -neighborhood of in . is uniformly continuous, for ; there exists such that implies for all , . Suppose that is an -ergodic pseudoorbit for ; let and . If , then . This implies that , and therefor is a -ergodic pseudoorbit for . Since has ergodic shadowing property, there exists in with . If , then , and we get . Now it is easy using definition to see that has ergodic shadowing property. Given by inverse shadowing property of there exists such that, for every and any -method for , there exists such that . Let = diam; choose such that . is a local homeomorphism; there exists with such that is a homeomorphism, where is the -neighborhood of in . is uniformly continuous, for ; there exists such that implies for all , . Let and let be a -method for . Define such that and ; we have . Therefor is a - method for , since has inverse shadowing property; thus for there exists such that ; this means that ; that is, has inverse shadowing property.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.