On homeomorphisms with the two-sided limit shadowing property

Abstract

We prove that the two-sided limit shadowing property is among the strongest known notions of pseudo-orbit tracing. It implies shadowing, average shadowing, asymptotic average shadowing and specification properties. We also introduce a weaker notion that is called two-sided limit shadowing with a gap and prove that it implies shadowing and transitivity. We show that those two properties allow to characterize topological transitivity and mixing in a class of expansive homeomorphisms and hence they characterize transitive (mixing) shifts of finite type.

Figures (1)

• Figure 1: A graph presenting the shift space $X_{(3,4)}$, given by the set $\mathcal{L}=\{01,12,20,03,34,45,50\}$.

Theorems & Definitions (29)

• Lemma 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• Theorem 3.4: KKO
• proof
• Theorem A
• proof