Birkhoff sum fluctuations in substitution dynamical systems

TL;DR

This paper studies the fluctuations of Birkhoff sums in substitution dynamical systems, establishing distributional limits and a new criterion for coboundaries, advancing understanding of their statistical properties.

Contribution

It introduces new results on the distributional convergence of Birkhoff sums and provides criteria for coboundaries in substitution systems, including a characterization of systems with bounded discrepancy.

Findings

Distributional convergence for Birkhoff sums of eigenfunctions

Central limit theorem for noncoboundary eigenfunctions with eigenvalue modulus 1

Characterization of substitution systems with bounded discrepancy

Abstract

We consider the deviation of Birkhoff sums along fixed orbits of substitution dynamical systems. We show distributional convergence for the Birkhoff sums of eigenfunctions of the substitution matrix. For noncoboundary eigenfunctions with eigenvalue of modulus 1, we obtain a central limit theorem. For other eigenfunctions, we show convergence to distributions supported on Cantor sets. We also give a new criterion for such an eigenfunction to be a coboundary, as well as a new characterization of substitution dynamical systems with bounded discrepancy