Exact invariant measures: How the strength of measure settles the intensity of chaos

TL;DR

This paper explores how the strength of invariant measures influences the intensity of chaos in dynamical systems, providing methods to identify ergodic properties and solutions for maps with known infinite measures.

Contribution

It introduces an approach to solve the inverse problem for nonlinear maps from invariant measures and links measure strength to chaos intensity, including solutions for maps with known infinite measures.

Findings

Infinite measures imply weak chaos with subexponential Lyapunov instability.

Finite measures correspond to fully chaotic maps with exponential instability.

Infinite measures lead to universal Mittag-Leffler statistical behavior.

Abstract

The aim of this paper is to show how extracting dynamical behavior and ergodic properties from deterministic chaos with the assistance of exact invariant measures. On the one hand, we provide an approach to deal with the inverse problem of finding nonlinear interval maps from a given invariant measure. Then, we show how to identify ergodic properties by means of transitions along the phase space via exact measures. On the other hand, we discuss quantitatively how infinite measures imply maps having subexponential Lyapunov instability (weakly chaotic), as opposed to finite measure ergodic maps, that are fully chaotic. In addition, we provide general solutions of maps for which infinite invariant measures are exactly known throughout the interval (a demand from this field). Finally, we give a simple proof that infinite measure implies universal Mittag-Leffler statistics of observables, rather than narrow distributions typically observed in finite measure ergodic maps.