Limit theorems for self-similar tilings

TL;DR

This paper investigates the deviation of ergodic averages in self-similar tiling dynamical systems, introducing finitely-additive measures and deriving limit theorems that describe their asymptotic behavior.

Contribution

It introduces a new family of finitely-additive measures and provides asymptotic formulas for ergodic integrals in self-similar tiling systems, leading to novel limit theorems.

Findings

Asymptotic formulas for ergodic integrals

Limit theorems for self-similar tilings

Introduction of finitely-additive measures

Abstract

We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems given by self-similar tilings.