Limit directions of a vector cocycle, remarks and examples

TL;DR

This paper investigates the set of limit directions of vector cocycles in dynamical systems, providing general results and specific models, with applications to billiard systems and cone sojourns.

Contribution

It introduces a comprehensive analysis of limit directions of vector cocycles, including descriptions for particular models and the behavior of cocycles in cones under invariance principles.

Findings

Characterization of limit directions in general and specific models

Description of limit directions in billiard systems with periodic obstacles

Analysis of cocycle behavior in cones under invariance principles

Abstract

We study the set ${\cal D}(\Phi)$ of limit directions of a vector cocycle $(\Phi_n)$ over a dynamical system, i.e., the set of limit values of $\Phi_n(x) /\|\Phi_n(x)\|$ along subsequences such that $\|\Phi_n(x)\|$ tends to $\infty$. This notion is natural in geometrical models of dynamical systems where the phase space is fibred over a basis with fibers isomorphic to $\mathbb{R}^d$, like systems associated to the billiard in the plane with periodic obstacles. It has a meaning for transient or recurrent cocycles. Our aim is to present some results in a general context as well as for specific models for which the set of limit directions can be described. In particular we study the related question of sojourn in cones of the cocycle when the invariance principle is satisfied.