Entropy conjugacy for Markov multi-maps of the interval

James P. Kelly; Kevin McGoff

doi:10.3934/dcds.2020353

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2021, Volume 41, Issue 5: 2071-2094. Doi: 10.3934/dcds.2020353

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Entropy conjugacy for Markov multi-maps of the interval

James P. Kelly^1, and
Kevin McGoff^2, ,

1.
Department of Mathematics, Christopher Newport University, Newport News, VA 23606, USA
2.
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

^* Corresponding author: Kevin McGoff
^* Corresponding author: Kevin McGoff

Received: September 30, 2019

Revised: May 31, 2020

Early access: October 2020

Published: May 2021

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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} $.

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Mathematics Subject Classification: Primary: 37B40; Secondary: 37B10, 54C60.

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Figure 1. The graph of a Markov multi-map and its corresponding adjacency matrix

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Figure 2. Markov multi-map from Example 9.1

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Figure 3. Markov multi-maps from Example 10.2 (left) and Example 10.3 (right)

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