On the completely irregular set for systems with the shadowing property

TL;DR

This paper demonstrates that for certain dynamical systems with the shadowing property, the set of completely irregular points is large and dense, revealing deep connections between transitivity, shadowing, and irregular behavior.

Contribution

It establishes the Baire genericity and density of completely irregular points in systems with shadowing and transitivity, extending understanding of irregular sets in dynamical systems.

Findings

Completely irregular set is Baire generic under specified conditions.

Orbit of any completely irregular point is dense.

Connections between transitivity and shadowing are analyzed.

Abstract

We prove that the completely irregular set is Baire generic for every non-uniquely ergodic transitive continuous map which satisfies the shadowing property and acts on a compact metric space without isolated points. We also show that, under the previous assumptions, the orbit of any completely irregular point is dense. Afterwards, we analyze the connection between transitivity and the shadowing property, draw a few consequences of their joint action within the family of expansive homeomorphisms, and discuss several examples to test the scope of our results.