Topological entropy of Markov set-valued functions

Abstract

We investigate the entropy for a class of upper semi-continuous set-valued functions, called Markov set-valued functions, that are a generalization of single-valued Markov interval functions. It is known that the entropy of a Markov interval function can be found by calculating the entropy of an associated shift of finite type. In this paper, we construct a similar shift of finite type for Markov set-valued functions and use this shift space to find upper and lower bounds on the entropy of the set-valued function.

Entropy conjugacy for Markov multi-maps of the interval

“…We now give a precise definition of Markov multi-maps on the interval [0, 1]. This definition is based on the one given in [2], though our definition is slightly less general. 1] , and for each a ∈ A, there exists p i ∈ P such that 1] , and for each a ∈ A, there exists u Our results require that F has some additional structure, which we now begin to define.…”

“…Recent work [6,5,12,3] has generalized the notion of Markov interval maps to the setting of multi-maps and established some of their basic properties. In particular, [2] proves that under some conditions on the Markov multi-map, one may find upper and lower bounds for its entropy using associated shifts of finite type.…”

Topological dynamics of Markov multi-maps of the interval

“…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].…”

Ivanov’s Theorem for Admissible Pairs Applicable to Impulsive Differential Equations and Inclusions on Tori

“…On the other hand, although there exist several various definitions of a topological entropy for multivalued maps (see e.g., [16][17][18][19][20][21][22]), as far as we know, none have been applied to differential equations without uniqueness or, more generally to differential inclusions. Moreover, no analogy of Theorem 1 exists for multivalued maps.…”