Making rare events typical in $d$-dimensional chaotic maps

TL;DR

This paper extends a large-deviation framework to $d$-dimensional chaotic maps, proposing a method to find effective maps that make rare events typical, demonstrated on 2D chaos examples.

Contribution

It introduces a method to identify effective topologically-conjugate maps for $d$-dimensional chaotic systems, enabling the study of rare events as typical.

Findings

Effective maps reproduce rare-event statistics in long-time limit

Method applied successfully to 2D tent map and Arnold's cat map

Framework generalizes to higher-dimensional chaotic systems

Abstract

Due to the deterministic nature of chaotic systems, fluctuations in their trajectories arise solely from the choice of initial conditions. Some of these dynamical fluctuations may lead to extremely unlikely scenarios. Understanding the impact of such rare events and the trajectories that give rise to them is of significant interest across disciplines. Yet, identifying the initial conditions responsible for those events is a challenging task due to the inherent sensitivity to small perturbations of chaotic dynamics. In a recent paper [Phys. Rev. Lett. 131, 227201 (2023)], this challenge was addressed by finding the effective dynamics that make rare events typical for one-dimensional chaotic maps. Here we extend such large-deviation framework to $d$-dimensional chaotic maps. Specifically, for any such map, we propose a method to find an effective topologically-conjugate map which reproduces the rare-event statistics in the long-time limit. We demonstrate the applicability of this result using several observables of paradigmatic examples of two-dimensional chaos, namely the two-dimensional tent map and Arnold's cat map.