Oscillating statistics of transitive dynamics

TL;DR

This paper proves that in certain dynamical systems, most orbits display highly oscillating statistical behavior, with their asymptotic statistics encompassing all ergodic measures, especially when the system is Lebesgue-ergodic.

Contribution

It establishes the generic occurrence of extremely oscillating asymptotic statistics in transitive, non-uniquely ergodic systems, expanding understanding of statistical complexity in dynamical systems.

Findings

Most orbits have asymptotic statistics containing all ergodic measures.

In Lebesgue-ergodic systems, almost all orbits exhibit oscillating statistics.

The minimal invariant measure set describes the asymptotic behavior of generic orbits.

Abstract

We prove that topologically generic orbits of C0 transitive and non-uniquely ergodic dynamical systems, exhibit an extremely oscillating asymptotical statistics. Precisely, the minimum weak* compact set of invariant probabilities, that describes the asymptotical statistics of each orbit of a residual set, contains all the ergodic probabilities. If besides f is ergodic with respect to the Lebesgue measure, then also Lebesgue-almost all the orbits exhibit that kind of extremely oscillating statistics.