Convergence to a L\'evy process in the Skorohod $M_1$ and $M_2$ topologies for nonuniformly hyperbolic systems, including billiards with cusps

TL;DR

This paper establishes conditions under which certain nonuniformly hyperbolic systems, including billiards with cusps, converge to Levy processes in specialized Skorohod topologies, extending previous stable law results.

Contribution

It provides geometric criteria for convergence to Levy processes in M_1 and M_2 topologies, broadening the understanding of limit laws for complex dynamical systems.

Findings

Convergence in M_1 topology under specific geometric conditions.

Convergence in M_2 topology but not in M_1 for certain systems.

Framework to derive functional limit laws from stable law convergence.

Abstract

We prove convergence to a Levy process for a class of dispersing billiards with cusps. For such examples, convergence to a stable law was proved by Jung & Zhang. For the corresponding functional limit law, convergence is not possible in the usual Skorohod J_1 topology. Our main results yield elementary geometric conditions for convergence (i) in M_1, (ii) in M_2 but not M_1. In general, we show for a large class of nonuniformly hyperbolic systems how to deduce functional limit laws once convergence to the corresponding stable law is known.