Shadowing, internal chain transitivity and α-limit sets

Abstract

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When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

“…In Section 3, we answer questions 1.1 and 1.2. Throughout, we present examples which serve both to motivate our results and also to demonstrate the distinction between some of the properties introduced in [21], [23] and in this paper. In particular, we construct an example of a system (Example 3.1) which demonstrates that one can have α f = ω f = ICT f whilst exhibiting neither property P e nor property P a .…”

“…Various approaches to this have been taken [2,16,17,27,45,46]. For a discussion on this, we refer the reader to [23]. In the present paper, we refrain from defining such sets for individual points, choosing instead to define them for backward and full trajectories.…”

“…In the present paper, we refrain from defining such sets for individual points, choosing instead to define them for backward and full trajectories. This is the approach taken in [2], [23] and [28]. An infinite sequence x i i≤0 is called a backward trajectory of x if f (x i ) = x i+1 for all i ≤ −1 and x 0 = x.…”

“…We define the limit sets of such a sequence in the natural way: α f ( x i i∈Z ) = α f ( x i i≤0 ) and ω f ( x i i∈Z ) = ω f (x 0 ). Although α-limit sets have not been studied quite as extensively as their ω counterparts, interest in them has been growing (see, for example, [2,16,17,23,27,28]).…”

On inverse shadowing

“…For example, Bowen [6] used shadowing implicitly as a key step in his proof that the nonwandering set of an Axiom A diffeomorphism is a factor of a shift of finite type. Since then it has been studied extensively as a key factor in stability theory [32,34,37], in understanding the structure of ω-limit sets and Julia sets [2,3,4,5,7,17,25], and as a property in and of itself [11,13,15,16,23,26,27,31,32,36].…”