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

Roberto Venegeroles
Phys. Rev. E 91, 062914 – Published 22 June 2015

Abstract

The aim of this paper is to show how to extract 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, which 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.

  • Figure
  • Received 10 March 2015

DOI:https://doi.org/10.1103/PhysRevE.91.062914

©2015 American Physical Society

Authors & Affiliations

Roberto Venegeroles*

  • Centro de Matemática, Computação e Cognição, UFABC, 09210-170, Santo André, SP, Brazil

  • *roberto.venegeroles@ufabc.edu.br

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Issue

Vol. 91, Iss. 6 — June 2015

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