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Two conjectures on coarse conjugacy by Geller and Misiurewicz

Damian Sawicki

TL;DR

The paper investigates two GM conjectures on coarse conjugacy within the framework of coarse entropy and large-scale dynamics. It constructs explicit counterexamples showing that coarse conjugacy need not carry to powers ($f^n$ and $g^n$) for all $n$, even when intertwinings via a coarse equivalence hold, and demonstrates that the conjecture about mutual coarse realizations can fail in the absence of extra hypotheses. A key positive result is that if the intertwining map $\varphi$ is surjective, then the conjecture holds: $f$ and $g$ are coarsely conjugate via $\varphi$ and a suitable coarse inverse $\psi$ with $\varphi\circ\psi=\mathrm{id}_Y$, whereas injective cases can fail. The work also shows the optimality of certain GM results under additional assumptions and delineates the boundary between dynamics-focused and map-focused hypotheses in coarse geometry. $

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.

Two conjectures on coarse conjugacy by Geller and Misiurewicz

TL;DR

The paper investigates two GM conjectures on coarse conjugacy within the framework of coarse entropy and large-scale dynamics. It constructs explicit counterexamples showing that coarse conjugacy need not carry to powers ( and ) for all , even when intertwinings via a coarse equivalence hold, and demonstrates that the conjecture about mutual coarse realizations can fail in the absence of extra hypotheses. A key positive result is that if the intertwining map is surjective, then the conjecture holds: and are coarsely conjugate via and a suitable coarse inverse with , whereas injective cases can fail. The work also shows the optimality of certain GM results under additional assumptions and delineates the boundary between dynamics-focused and map-focused hypotheses in coarse geometry. $

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.
Paper Structure (5 sections, 15 theorems, 48 equations)

This paper contains 5 sections, 15 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Counterexamples to Conjecture \ref{['conjecture2']}
  3. A true version of and a counterexample to Conjecture \ref{['conjecture']}
  4. Surjectivity of conjugating maps
  5. Counterexample

Key Result

Theorem 1.2

GM*Proposition 2.8 and Corollary 2.13 If $f$ and $g$ are coarsely conjugate and $g$ is controlled, then $f$ is controlled as well and for any natural $n$ the maps $f^n$ and $g^n$ are coarsely conjugate via the same coarse equivalences as $f$ and $g$.

Theorems & Definitions (35)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1
  • Theorem 1.3
  • Conjecture 2
  • Theorem 3
  • Example 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • ...and 25 more