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

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.

20 pages, 15 figures. Accepted for publication in Phys. Rev. E. © 2026 American Physical Society. This is the accepted manuscript of the corresponding journal reference available at (external doi)

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