Measurable Time-Restricted Sensitivity

TL;DR

This paper introduces two new concepts of time-restricted sensitivity in measurable dynamical systems, linking them to metric entropy and exploring their behavior in various transformation classes.

Contribution

It defines novel notions of time-restricted sensitivity and relates them to metric entropy, extending analysis to non-measure-preserving transformations.

Findings

Time before divergence is logarithmic in initial distance measure.

Relations established between sensitivity notions and metric entropy.

Analysis of sensitivity in non-measure-preserving transformations.

Abstract

We develop two notions of time-restricted sensitivity to initial conditions for measurable dynamical systems, where the time before divergence of a pair of paths is at most an asymptotically logarithmic function of a measure of their initial distance. In the context of finite measure-preserving transformations on a compact space, we relate these notions to the metric entropy of the system. We examine one of these notions for classes of non-measure-preserving, nonsingular transformations.