Two conjectures on coarse conjugacy by Geller and Misiurewicz

TL;DR

This paper disproves two conjectures on coarse conjugacy introduced by Geller and Misiurewicz, analyzing the conditions under which these conjectures hold or fail in the context of coarse entropy in metric space dynamics.

Contribution

The paper provides counterexamples to both conjectures and explores the effects of additional assumptions on their validity, refining understanding of coarse conjugacy.

Findings

Disproved both conjectures on coarse conjugacy.

Identified optimality conditions for Geller and Misiurewicz's results.

Proved one conjecture under a new, complementary assumption.

Abstract

In their study of coarse entropy, W. Geller and M. Misiurewicz introduced the notion of coarse conjugacy: a version of conjugacy appropriate for dynamics on metric spaces observed from afar. They made two conjectures on coarse conjugacy generalising their results. We disprove both of these conjectures. We investigate the impact of extra assumptions on the validity of the conjectures: We show that the result of Geller and Misiurewicz towards one of the conjectures can be considered optimal, and we prove the other conjecture under an assumption complementary to that from the referenced work.