Spatial Recurrence for Ergodic Fractal Measures

TL;DR

This paper explores the regularity and ergodic properties of measures under spatial recurrence, revealing how measure translation actions behave non-singularly and establishing key ergodic theorems for these systems.

Contribution

It introduces an invertible framework for Furstenberg's ergodic systems, linking measure magnification regularity to translation regularity, and proves new ergodic theorems for measure translation actions.

Findings

Translation action on measures is non-singular.

Established pointwise discrete ergodic theorem.

Proved continuous ergodic theorem for measure translation.

Abstract

We discuss an invertible version of Furstenberg's `Ergodic CP Shift Systems'. We show that the explicit regularity of these dynamical systems with respect to magnification of measures, implies certain regularity with respect to translation of measures; We show that the translation action on measures is non-singular, and prove pointwise discrete and continuous ergodic theorems for the translation action.