Birkhoff sums as distributions I: Regularity

TL;DR

This paper investigates the regularity properties of Birkhoff sums viewed as distributions across different hyperbolic dynamical systems, with applications to advection in discrete systems.

Contribution

It provides new regularity results for Birkhoff sums as distributions in hyperbolic dynamical systems, extending understanding of their behavior.

Findings

Regularity results for Birkhoff sums in hyperbolic systems

Application to advection in discrete dynamical systems

Analysis of distributions for linear and piecewise expanding maps

Abstract

We study Birkhoff sums as distributions. We obtain regularity results on such distributions for various dynamical systems with hyperbolicity, as hyperbolic linear maps on the torus and piecewise expanding maps on the interval. We also give some applications, as the study of advection in discrete dynamical systems.