Infinite Ergodicity that Preserves the Lebesgue Measure

TL;DR

This paper demonstrates that a class of infinite one-dimensional dynamical systems preserve Lebesgue measure, are ergodic, and exhibit weak chaos, with Lyapunov exponents following a Mittag-Leffler distribution, extending prior foundational work.

Contribution

It generalizes previous results by showing infinite ergodicity and measure preservation for a broad class of parameterized systems, connecting exact and dissipative regimes.

Findings

Systems preserve Lebesgue measure and are ergodic.

Lyapunov exponents follow Mittag-Leffler distribution of order 1/2.

Results extend classical work by Adler and Weiss.

Abstract

We proved that for the countably infinite number of one-parameterized one dimensional dynamical systems, they preserve the Lebesgue measure and they are ergodic for the measure (infinite ergodicity). Considered systems connect the parameter region in which dynamical systems are exact and the parameter region in which systems are dissipative, and correspond to the critical points of the parameter in which weak chaos occurs (the Lyapunov exponent converges to zero). These results are the generalization of the work by R. Adler and B. Weiss. We show that the distributions of normalized Lyapunov exponent for these systems obey the Mittag-Leffler distribution of order $1/2$ by numerical simulation.