Expanding and expansive time-dependent dynamics

TL;DR

This paper investigates time-dependent dynamical systems with expanding maps, deriving entropy formulas, establishing conjugacy to autonomous systems under certain conditions, and introducing strong uniform expansivity to generalize classical results.

Contribution

It provides entropy formulas for expanding systems, shows conjugacy to autonomous systems under C^1 conditions, and introduces strong uniform expansivity for time-dependent dynamics.

Findings

Derived formulas for metric and topological entropy.

Proved existence of equi-conjugacy to autonomous systems under C^1 assumptions.

Generalized classical results using strong uniform expansivity.

Abstract

In this paper, time-dependent dynamical systems given by sequences of maps are studied. For systems built from expanding C^2-maps on a compact Riemannian manifold M with uniform bounds on expansion factors and derivatives, we provide formulas for the metric and topological entropy. If we only assume that the maps are C^1, but act in the same way on the fundamental group of M, we can show the existence of an equi-conjugacy to an autonomous system, implying a full variational principle for the entropy. Finally, we introduce the notion of strong uniform expansivity that generalizes the classical notion of positive expansivity, and we prove time-dependent analogues of some well-known results. In particular, we generalize Reddy's result which states that a positively expansive system locally expands distances in an equivalent metric.