Invariance entropy for topological semigroup actions (Q2855921)

Let \(S\) be a topological semigroup with the identity \(e\) acting on a metric space \(M\). Assume that there exists a family \(A_{\tau} \subset S\), \(\tau \in [0, \infty)\), such that \(e \in A_{\tau}\) and \(A_{\tau} A_{\sigma} \subset A_{\tau+\sigma}\) for every \(\tau, \sigma \geq 0\). Under these conditions, the authors introduce the property of being weakly almost invariant for subsets \(Q \subset M\) and the notion of invariance entropy for compact subsets of weakly almost invariant sets. They obtain an upper and a lower bound for invariance entropy. The authors show that the invariance entropy introduced in the present paper coincides with the invariance entropy defined in [\textit{F. Colonius} and \textit{C. Kawan}, SIAM J. Control Optim. 48, No. 3, 1701--1721 (2009; Zbl 1193.94049)] for controlled almost invariant sets considered in the context of control systems.