(Non)Invariance of dynamical quantities for orbit equivalent flows

TL;DR

This paper investigates how key dynamical quantities change under time reparameterizations in dynamical systems, revealing conditions for invariance and transformation, with implications for chaos in relativity and state synchronization.

Contribution

It provides a detailed analysis of the invariance and transformation rules of dynamical quantities under time reparameterizations, highlighting their significance in physical and mathematical contexts.

Findings

Dynamical quantities can be invariant, multiplicatively transformed, or depend on integrals of initial conditions.

Results have implications for understanding chaos invariance in general relativity.

Applications include synchronization of equilibrium states and controlling expansions.

Abstract

We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent non-invariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions.