Every expanding measure has the nonuniform specification property

TL;DR

This paper proves that any expanding measure with positive Lyapunov exponents satisfies the nonuniform specification property, providing new insights into recurrence and periodic approximation in dynamical systems.

Contribution

It establishes that all expanding measures have the nonuniform specification property, linking hyperbolic times, recurrence, and periodic points in a novel way.

Findings

Expanding measures satisfy the nonuniform specification property.

Periodic points of sublinear period growth are dense in measure.

Recurrence estimates relate to Lyapunov exponents.

Abstract

Exploring abundance and non lacunarity of hyperbolic times for endomorphisms preserving an ergodic probability with positive Lyapunov exponents, we obtain that there are periodic points of period growing sublinearly with respect to the lenght of almost every dynamical ball. In particular, we conclude that any ergodic measure with positive Lyapunov exponents satisfy the nonuniform specification property. As consequences, we (re)-obtain estimates on the recurrence to a ball in terms of the Lyapunov exponents and we prove that any expanding measure is limit of Dirac measures on periodic points.