Local Shadowing and Entropy for Homeomorphisms

TL;DR

This paper explores how local shadowing influences the entropy and measure approximation in homeomorphisms, revealing new insights into local versus global shadowing phenomena and constructing novel examples.

Contribution

It introduces a method to approximate invariant measures with ergodic measures of higher entropy using local shadowing, and distinguishes local shadowing from global shadowing through new examples.

Findings

Local shadowing allows approximation of measures with higher entropy.

Merging dynamical properties can induce positive entropy.

New examples show local shadowing differs from global shadowing.

Abstract

In this work, we investigate the dynamics of homeomorphisms through the lens of the local shadowing theory. We study the influence of positively shadowable points and positively shadowable measures into the local entropy theory of homeomrphisms. Specifically, we use pointwise shadowing to approximate invariant measures by ergodic measures with bigger entropy and supported on semi-horseshoes, giving certain flexibility of entropy to systems with local shadowing. We further apply our findings to show that by merging some pointwise dynamical properties, one can lead to positive entropy, significantly enhancing several related results previously established in the field. Besides, we introduce new examples of dynamical systems with local shadowing phenomena, by providing a way of construct examples, but lacking of several properties present in systems with the global shadowing, showing that the local and global shadowing are, in fact, distinct phenomena.