Shadowing, entropy and minimal subsystems

Abstract

We consider non-wandering dynamical systems having the shadowing property, mainly in the presence of sensitivity or transitivity, and investigate how closely such systems resemble the shift dynamical system in the richness of various types of minimal subsystems. In our excavation, we do discover regularly recurrent points, sensitive almost 1-1 extensions of odometers, minimal systems with positive topological entropy, etc. We also show that transitive semi-distal systems with shadowing are in fact minimal equicontinuous systems (hence with zero entropy) and, in contrast to systems with shadowing, the entropy points do not have to be densely distributed in transitive systems.

On the irregular points for systems with the shadowing property

“…Now we are ready to prove our first main result. It was proved in [39] that if a dynamical system with the shadowing property has a recurrent not minimal point, or a minimal sensitive point, then the entropy must be positive. A similar analysis will be performed here.…”

Positive entropy through pointwise dynamics

“…In this section we will give a proof for Theorem 1. We start stating a result due to Kawaguchi, N. which exhibits an abstraction of the techniques developed in [13] by Moothathu and Oprocha. Lemma 6 (Lemma 2.3 in [5]).…”

“…Using the above concepts and some methods developed in [13] we obtain our main result which can be stated as follows. Theorem 1.…”

Shadowing Property, Weak Mixing and Regular Recurrence

“…In [16] Moothathu proved that in every non-wandering dynamical system with the shadowing property the set of minimal points is dense and recently this result was extended in [17], by showing that the set of regularly recurrent points is also dense. If a non-wandering dynamical system with the shadowing property is sensitive, various types of minimal subsystems are present in the dynamics, such as sensitive almost 1-1 extension of odometers and minimal subsystems with positive entropy.…”