Remarkable universalities in distributions of dynamical observables in chaotic systems

The study of chaotic systems, where rare events play a pivotal role, is essential for understanding complex dynamics due to their sensitivity to initial conditions. Recently, tools from large deviation theory, typically applied in the context of stochastic processes, have been used in the study of chaotic systems. Here, we study dynamical observables, $A = \sum_{n=1}^N g(\textbf{x}_n)$, defined along a chaotic trajectory $\{\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_N\}$. For most choices of $g(\textbf{x})$, $A$ satisfies a central limit theorem: At large sequence size $N \gg 1$, typical fluctuations of $A$ follow a Gaussian distribution with a variance that scales linearly with $N$. Large deviations of $A$ are usually described by the large deviation principle, that is, $P(A) \sim e^{- N I(A/N)}$, where $I(a)$ is the rate function. We find that certain dynamical observables exhibit a remarkable statistical universality: even when constructed with distinct functions $g_1(\textbf{x})$ and $g_2(\textbf{x})$, different observables are described by the same rate function. We provide a physical interpretation for this striking universality by showing that $g_1(\textbf{x})-g_2(\textbf{x})$ belongs to a class of functions that we call ``derived''. Furthermore, we show that if $g(\textbf{x})$ itself is ``derived'', then the distribution of $A$ becomes independent of $N$ in the large-$N$ limit, and is generally non-Gaussian (although it is mirror-symmetric). We demonstrate that the finite-time Lyapunov exponent (FTLE) for the logistic map is of this derived form, thus providing a simple explanation for some existing results.