Stable laws for heavy-tailed observables on polynomially mixing billiards

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 $(\frac{1}{\alpha}+\gamma)^{-1}$.