Entropy conjugacy for Markov multi-maps of the interval

Abstract

We consider a class F of Markov multi-maps on the unit interval. Any multi-map gives rise to a space of trajectories, which is a closed, shiftinvariant subset of [0, 1] Z +. For a multi-map in 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 F .

Topological dynamics of Markov multi-maps of the interval

“…Throughout the paper we assume that all Markov multi-maps are properly parametrized. (As was shown in [18], if F 0 is any Markov multi-map, there exists a properly parametrized Markov multi-map F 1 such that G(F 0 ) = G(F 1 ).) We also note that Section 8 contains two examples of properly parametrized Markov multi-maps.…”

“…In this context, one may associate to any Markov multi-map of the interval a shift of finite type (SFT) that captures the combinatorics of the multi-map. Markov multi-maps have been studied in recent years with a focus on how the associated SFT can be used to investigate the topological structure of the inverse limit [2,5,6,11], as well as its topological entropy [3,18].…”