Invariance entropy for topological semigroup actions

Abstract

Abstract. Invariance entropy for the action of topological semigroups acting on metric spaces is introduced. It is shown that invariance entropy is invariant under conjugations and a lower bound and upper bounds of invariance entropy are obtained. The special case of control systems is discussed.

Invariance Pressure for Control Systems

“…If S ⊂ U is a strongly (nN, Q)-spanning set for (1), then S N := {i(ω); ω ∈ S} is a strongly (n, Q)-spanning set for (11). Analogously, if S N is a strongly (n, Q)-spanning set for (11), then i −1 (S N ) is a strongly (nN, Q)spanning set for (1). Therefore…”

“…Basic references are the seminal paper Nair, Evans, Mareels and Moran [7] and the monograph Kawan [6]. Further studies of invariance entropy include Da Silva and Kawan [4] for hyperbolic control sets, Da Silva [3] for linear control systems on Lie groups and Colonius, Fukuoka and Santana [1] for topological semigroups.…”

Entropy for semigroup actions on general topological spaces

“…We can also establish a relationship between topological entropy and invariance entropy. On the other hand, in the paper [10] the authors consider a notion of invariance entropy not in the context of cocycle on topological spaces, but looking at actions of semigroups endowed with admissible families. In this setting, a semigroup S acting on a metric space M is required to admit admissible families associated to a given regular system in S .…”

“…In this setting, a semigroup S acting on a metric space M is required to admit admissible families associated to a given regular system in S . The notion of weak almost invariant set depends on the admissible families, and thus the invariance entropy is determined by the effects of the admissible families on the action of the semigroup ([10, Defs. 1.7 and 1.10]).…”

“…Thus considering the importance of dynamic properties in the study of semigroup actions, it is natural to introduce concepts of entropy in the general theory of topological dynamics. From invariance entropy for control systems, in 2013, Colonius, Fukuoka, and Santana [10] introduced the notion of invariance entropy for semigroup actions on metric spaces. In 2014, Zhou, Zhang, and Chen [39], inspired by the topological conditional entropy defined by Misiurewicz, presented the topological conditional entropy for countable discrete amenable group actions on compact metric spaces.…”