Time-metric equivalence and dimension change under time reparameterizations

TL;DR

This paper investigates how time reparameterizations affect dynamical systems, revealing their equivalence to metric transformations and analyzing the invariance of Lyapunov exponents and generalized dimensions, with implications for chaos characterization.

Contribution

It establishes the local equivalence of time and metric transformations and analyzes the invariance of generalized dimensions under time reparameterizations.

Findings

Lyapunov exponents transform according to a specific rule under time reparameterizations.

Generalized dimensions D_q are preserved except for D_1, which transforms nontrivially.

The behavior at q=1 provides insights into the Kaplan-Yorke conjecture.

Abstract

We study the behavior of dynamical systems under time reparameterizations, which is important not only to characterize chaos in relativistic systems but also to probe the invariance of dynamical quantities. We first show that time transformations are locally equivalent to metric transformations, a result that leads to a transformation rule for all Lyapunov exponents on arbitrary Riemannian phase spaces. We then show that time transformations preserve the spectrum of generalized dimensions D_q except for the information dimension D_1, which, interestingly, transforms in a nontrivial way despite previous assertions of invariance. The discontinuous behavior at q=1 can be used to constrain and extend the formulation of the Kaplan-Yorke conjecture.