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

TL;DR

This paper proves Barge's entropy conjecture by constructing pseudo-arc homeomorphisms with any specified entropy value between 0 and infinity, filling a longstanding gap in topological dynamics.

Contribution

It provides the first construction of pseudo-arc homeomorphisms with arbitrary finite and infinite entropy, advancing understanding of entropy spectrum in topological dynamics.

Findings

Constructed pseudo-arc homeomorphisms with any entropy in [0, ∞]

Resolved Barge's 1989 entropy conjecture

Extended the known entropy spectrum for pseudo-arc homeomorphisms

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 $\infty$.