Expanding measure has nonuniform specification property on random dynamical system

TL;DR

This paper investigates the nonuniform specification property in random dynamical systems, demonstrating how ergodic measures with positive Lyapunov exponents relate to return points and hyperbolic times along fibers.

Contribution

It establishes the nonuniform specification property for ergodic measures with positive Lyapunov exponents in RDS and analyzes the distribution of return points and hyperbolic times.

Findings

Ergodic measures with positive Lyapunov exponents satisfy the nonuniform specification property.

Expanding measures can be approximated by measures supported on finitely many return points.

Average measures along orbits converge to Dirac measures supported on return orbits.

Abstract

In the present paper, we study the distribution of the return points in the fibers for a RDS (random dynamical systems) nonuniformly expanding preserving an ergodic probability, we also show the abundance of nonlacunarity of hyperbolic times that are obtained along the orbits through the fibers. We conclude that any ergodic measure with positive Lyapunov exponents satisfies the nonuniform specification property between fibers. As consequences, we prove that any expanding measure is the limit of probability measure whose measures of disintegration on the fibers are supported by a finite number of return points and we prove that the average of the measures on the fibers corresponding to a disintegration, along an orbit in the base dynamics is the limit of Dirac measures supported in return orbits on the fibers.