# Decay of distance autocorrelation and Lyapunov exponents

## TL;DR

This paper investigates how the decay of distance autocorrelation in discrete dynamical systems relates to Lyapunov exponents, revealing different decay laws for various systems and linking decay exponents to Lyapunov exponents.

## Contribution

It provides numerical evidence connecting the decay of autocorrelation to Lyapunov exponents across different dynamical systems, highlighting system-specific decay laws.

## Key findings

- Exponential decay for quadratic map
- Logarithmic decay for Hénon map
- Power-law decay for standard map with decay exponent near Lyapunov exponent

## Abstract

This work presents numerical evidences that for discrete dynamical systems with one positive Lyapunov exponent the decay of the distance autocorrelation is always related to the Lyapunov exponent. Distinct decay laws for the distance autocorrelation are observed for different systems, namely exponential decays for the quadratic map, logarithmic for the H\'enon map and power-law for the conservative standard map. In all these cases the decay exponent is close to the positive Lyapunov exponent. For hyperbolic conservative systems, the power-law decay of the distance autocorrelation tends to be guided by the smallest Lyapunov exponent.