Uniform scaling limits for ergodic measures

TL;DR

This paper demonstrates that ergodic measures on one-sided shift spaces exhibit uniform scaling behavior, with scenery distributions converging to a common limit described explicitly via a reverse Jacobian function.

Contribution

It establishes uniform scaling limits for ergodic measures on shift spaces and provides an explicit description of the limiting distribution.

Findings

Scenery distributions converge weakly at almost every point.

The limiting distribution is explicitly characterized by a reverse Jacobian.

Uniform scaling behavior holds for ergodic measures on shift spaces.

Abstract

We prove that ergodic measures on one-sided shift spaces are uniformly scaling in the sense of Gavish. That is, given a shift ergodic measure we prove that at almost every point the scenery distributions weakly converge to a common distribution on the space of measures. Moreover, we give an explicit description of the limiting distribution in terms of a `reverse Jacobian' function associated with the corresponding measure on the space of left infinite sequences.