Second Order Ergodic Theorem for Self-Similar Tiling Systems

TL;DR

This paper proves a second-order ergodic theorem for self-similar tiling systems with infinite measure, linking the ergodic behavior to the Hausdorff dimension of an associated fractal set.

Contribution

It establishes the second-order ergodic theorem for a class of infinite measure self-similar tiling systems, connecting ergodic properties to fractal geometry.

Findings

Second-order ergodic theorem proven for self-similar tilings

Exponent matches Hausdorff dimension of a related fractal set

Extends ergodic theory to infinite measure tiling systems

Abstract

We consider infinite measure-preserving non-primitive self-similar tiling systems in Euclidean space $\mathbb R^d$. We establish the second-order ergodic theorem for such systems, with exponent equal to the Hausdorff dimension of a graph-directed self-similar set associated with the substitution rule.