The speed with which an orbit approaches a limit point

Positive topological entropy and Δ-weakly mixing sets

“…In this section, we study the non-classical Li-Yorke chaos. We will answer affirmatively three open questions stated in [33] and generalize some results obtained in [19,2] in this section. To do this we start from some notions.…”

“…In the final section we will discuss the non-classical Li-Yorke chaos, which generalizes several well known results in [2] and also answer several open questions in [33] (see Questions 6.1, 6.10 and 6.14 stated in our terminology).…”

“…Among other things, Moothathu [33] introduced a concept of non-classical Li-Yorke chaos. For two positive integers r, s ∈ N, we say that a pair (x, y) ∈ X × X is Similarly, we can define (r, s)-scrambled sets and (r, s)-Li-Yorke chaos.…”

“…Also in 2002, Blanchard, Glasner, Kolyada and Maass [6] proved that positive topological entropy also implies Li-Yorke chaos. Among other things, Moothathu [33] introduced a concept of non-classical Li-Yorke chaos. For two positive integers r, s ∈ N, we say that a pair (x, y) ∈ X × X is (r, s)scrambled if lim inf n→∞ ρ(T rn x, T sn y) = 0 and lim sup n→∞ ρ(T rn x, T sn y) > 0.…”

“…Let (X , T ) be a dynamical system which is totally transitive, proximal and uniformly rigid. It is shown in [33,Proposition 5] that if (X , T ) is proximal then Prox(T r , T s ) = X 2 for all r, s ∈ N. We also have Asy(T k , T k )\∆ X = / 0 for k ∈ N by the uniform rigidity. We want to show that Asy(T r , T r+s ) \ ∆ = / 0 for r, s ∈ N. Assume that (x, y) ∈ Asy(T r , T r+s ).…”