A type of shadowing and distributional chaos

Abstract

For any continuous self-map of a compact metric space, we extend a partition of each chain component with respect to a chain proximal relation to a G δ -partition of the phase space. Under the assumption of s-limit shadowing, we use this partition to give a global description of Li-Yorke type chaos corresponding to several Furstenberg families.

On -genericity of distributional chaos

“…Here, we have h top (f ) > 0 in both cases. This result is extended in [15] by using a relation defined by Richeson and Wiseman [33] (see also [14]). Note that it was previously known that shadowing with (chain) mixing implies the specification and so DC1 2 , except for the degenerate case (see [20,25,28]).…”

“…Since DC1 2 (X, f ) = ∅ implies the existence of a distal pair for f (see [29,Corollary 15]), any proximal map f ∈ C(X) with h top (f ) > 0, given in, for example, [12,19,30], does not exhibit DC1 2 . By [4,Theorem 2], we also know that a minimal map f ∈ C(X) with a regularly recurrent point satisfies DC1 2 (X, f ) = ∅, so every Toeplitz subshift with arbitrary topological entropy does not exhibit DC1 2 (see also [7] and [15,Remark 2.2]). Thus, some additional assumptions besides h top (f ) > 0 are needed to ensure DC1 n , n ≥ 2, for a general map f ∈ C(X).…”

“…The next lemma relates the previous lemma to a method developed in [15]. Let π = (π n+1 n : (X n+1 , f n+1 ) → (X n , f n )) n≥1 be an inverse sequence of equivariant maps with MLC(1) and let (X, f ) = lim π (X n , f n ).…”

“…To prove Lemma 1.3, we use the method in [15]. The next lemma is a modification of [15,Lemma 2.6]. LEMMA 5.8.…”

“…Most of this paper is devoted to a study of the MLC focusing on the structure of chain components. By using the above lemmas and a method in [15] with Mycielski's theorem, we prove the following lemma. Here, D(f ) is the partition of X with respect to the equivalence relation ∼ f defined by Richeson and Wiseman (see §2.2 for details).…”