Entropy conjugacy for Markov multi-maps of the interval

TL;DR

This paper studies Markov multi-maps on the interval, showing their trajectory spaces are entropy conjugate to shifts of finite type and characterizing possible topological entropies.

Contribution

It establishes entropy conjugacy for a class of Markov multi-maps and characterizes their topological entropy spectrum.

Findings

Trajectory spaces are entropy conjugate to shifts of finite type.

Characterization of attainable topological entropy values.

Provides a framework linking multi-maps to symbolic dynamics.

Abstract

We consider a class $\mathcal{F}$ of Markov multi-maps on the unit interval. Any multi-map gives rise to a space of trajectories, which is a closed, shift-invariant subset of $[0,1]^{\mathbb{Z}_+}$. For a multi-map in $\mathcal{F}$, we show that the space of trajectories is (Borel) entropy conjugate to an associated shift of finite type. Additionally, we characterize the set of numbers that can be obtained as the topological entropy of a multi-map in $\mathcal{F}$.