Topological entropy of compact subsystems of transitive real line maps

TL;DR

This paper investigates the minimal topological entropy of transitive maps on the real line and half-open interval, establishing a lower bound that is not attained, thus solving a previously posed problem.

Contribution

It proves the exact lower bounds of topological entropy for transitive real line maps, resolving a question about whether these bounds can be achieved.

Findings

Lower bound of ent(f) is log(√3) for maps on the real line.

Lower bound of ent(f) is log(3) for maps on [0,1).

These bounds are not attained by any transitive map.

Abstract

For a continuous map $f$ from the real line (half-open interval $[0,1)$) into itself let ent(f) denote the supremum of topological entropies of $f|_K$, where $K$ runs over all compact $f$-invariant subsets of $\mathbb{R}$ ($[0,1)$, respectively). It is proved that if $f$ is topologically transitive, then the best lower bound of ent(f) is $\log\sqrt{3}$ ($\log 3$, respectively) and it is not attained. This solves a problem posed by C{\'a}novas [Dyn. Syst. \textbf{24} (2009), no. 4, 473--483].