Stable laws for heavy-tailed observables on polynomially mixing billiards

TL;DR

This paper studies how slow mixing and heavy-tailed observables interact in polynomially mixing billiards, establishing stable limit laws depending on system parameters and extending results to intermittent maps.

Contribution

It demonstrates the interaction between slow mixing and heavy tails in billiards, deriving stable laws and extending previous results to broader parameter ranges.

Findings

Stable limit laws depend on the mixing exponent and tail index.

Interaction causes a transition between observable-driven and dynamics-driven stable laws.

Extension of stable law results to all parameters in certain intermittent maps.

Abstract

We investigate the competition between two distinct mechanisms generating stable laws in deterministic dynamical systems: slow mixing of the system and heavy-tailed observables. For heavy-tailed observables on polynomially mixing billiards with cusps we show these two mechanisms interact and there is a transition, depending on the mixing exponent and the index of the heavy-tailed observable, such that the limit law is determined by either the observable or the dynamics. We prove stable limit laws for heavy-tailed observables of the form $\phi(x)= d(x,x_0)^{-\frac{2}{\alpha}}, 0< \alpha < 2$, where $x_{0} \in \partial Q$ is a generic point on the dynamical system given by the collision map of a polynomially mixing billiard $(T, Q, \mu)$ with cusps. The observable $\phi$ has a tail of stable index $\alpha$, i.e. $\mu(|\phi|>t) \sim t^{-\alpha}$. The billiard systems we consider have a slow mixing rate so that suitably scaled H\"{o}lder observables on the billiard satisfy a stable law of index $1/\gamma$, with $\gamma$ a function of the flatness of the cusps. We establish stable limit laws satisfied by Birkhoff sums of $\phi$ for the parameter range $\gamma \in (1/2,1)$, $\alpha \in (0,2)$ ($\alpha \not =1$) as a function of $\gamma$ and $\alpha$. As an application, in the setting of intermittent maps, we extend the results of~\cite{CNT2025} to cover all parameter values of the map and the observable $\phi(x)= d(x,x_0)^{-\frac{1}{\alpha}}$ (which has stable index $\alpha$ if $x_0\not =0$) in the regime $0< \alpha < 2$, $0<\gamma<1$. We show if $x_0=0$, the indifferent fixed point, then the stable law has index $(\frac{1}{\alpha}+\gamma)^{-1}$.