Expansiveness, generators and Lyapunov exponents for finitely generated monoid actions

In this paper, we generalize the results of Fathi by establishing that a compact metrizable space admits an expansive finitely generated monoid action if, and only if, it possesses a compatible hyperbolic metric. Furthermore, we demonstrate the equivalence of the concepts of expansiveness, increasing small distances, and expanding small distances within a rather general framework. Additionally, we affirm that a compact metrizable space admits an expansive countable group action precisely when it has a generator. Moreover, we prove that the expansiveness property of group actions is inherited by finite-index subgroups and finite extensions. Lastly, we exhibit that the Lyapunov exponents for an expansive system are necessarily nonzero, thereby indicating that such a system exhibits chaotic behavior. Concurrently, we also demonstrate that negative Lyapunov exponents for compact invariant sets of a dynamical system imply that the compact set in question functions as an attractor.