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Beyond $0$ and $\infty$: A solution to the Barge Entropy Conjecture

Jan P. Boroński, Jernej Činč, Piotr Oprocha

TL;DR

The paper settles M. Barge's 1989 question by constructing, for every nonnegative entropy value $r$, a pseudo-arc homeomorphism with topological entropy $h_{top}=r$, using a Denjoy–Rees-type enrichment applied to an inverse-limit pseudo-arc with Cantor-fan dynamics. The approach combines inverse-limit techniques, crookedness control, and a carefully designed waste-bin scheme to manage invariant measures and realize prescribed entropy via the variational principle. This yields positive-entropy pseudo-arc dynamics without horseshoes and broadens the understanding of entropy rigidity vs. flexibility on one-dimensional continua. The work also illuminates the richness of pseudo-arc dynamics beyond interval dynamics and addresses related questions about periodic structure and homoclinic behavior.

Abstract

We prove the entropy conjecture of M. Barge from 1989: for every $r\in [0,\infty]$ there exists a pseudo-arc homeomorphism $h$, whose topological entropy is $r$. Until now all pseudo-arc homeomorphisms with known entropy have had entropy $0$ or $\infty$.

Beyond $0$ and $\infty$: A solution to the Barge Entropy Conjecture

TL;DR

The paper settles M. Barge's 1989 question by constructing, for every nonnegative entropy value , a pseudo-arc homeomorphism with topological entropy , using a Denjoy–Rees-type enrichment applied to an inverse-limit pseudo-arc with Cantor-fan dynamics. The approach combines inverse-limit techniques, crookedness control, and a carefully designed waste-bin scheme to manage invariant measures and realize prescribed entropy via the variational principle. This yields positive-entropy pseudo-arc dynamics without horseshoes and broadens the understanding of entropy rigidity vs. flexibility on one-dimensional continua. The work also illuminates the richness of pseudo-arc dynamics beyond interval dynamics and addresses related questions about periodic structure and homoclinic behavior.

Abstract

We prove the entropy conjecture of M. Barge from 1989: for every there exists a pseudo-arc homeomorphism , whose topological entropy is . Until now all pseudo-arc homeomorphisms with known entropy have had entropy or .

Paper Structure

This paper contains 13 sections, 27 theorems, 76 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Pseudo-arc: historic overview, characterizations and its appearance across various branches of mathematics
  4. Topological rigidity and flexibility
  5. Work leading to the solution of Barge's entropy conjecture
  6. Description of the main ingredients of the proof
  7. Preliminaries
  8. Crookedness Revisited
  9. Proof of Theorem \ref{['odoOld']}
  10. The Denjoy-Rees technique
  11. Proof of Theorem \ref{['thm:main']}
  12. Crucial properties.
  13. (by George Kozlowski${}^{5}$)

Key Result

Theorem 1.1

For every $r\in [0,\infty]$ there exists a pseudo-arc homeomorphism $H_r$ such that $h_{top}(H_r)=r$.

Figures (6)

  • Figure 1: Sketch of a graph of the adjusted map $h$ on $T_2$ from Lemma \ref{['tunc:3.3']}.
  • Figure 2: A sketch of how the graph of $f$ on $T^2$ is transformed to the graph of $\bar{f}|_{T_2}$ (left) and the smash map $\zeta$ (right).
  • Figure 3: Sketch of construction of near-homeomorphism $\bar{f}_{k,k+1}$ from the proof of Claim \ref{['clm:extension']}.
  • Figure 4: The idea of the construction of sets $\Lambda_n$ in Lemma \ref{['lem:setK']}.
  • Figure 5: The Denjoy-Rees-like enrichment on the pseudo-arc.
  • ...and 1 more figures

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • Definition 3.6
  • ...and 50 more