Distinguished correlation properties of Chebyshev dynamical systems and their generalisations

TL;DR

This paper investigates the unique correlation properties of Chebyshev maps and their generalizations, revealing their minimal higher-order correlations among conjugated maps, and explores their spectral characteristics and coupled map lattices.

Contribution

It provides analytic results for correlation functions of Chebyshev maps and their generalizations, and analyzes spectral properties and coupled systems.

Findings

Chebyshev maps have the least higher-order correlations among conjugated maps.

The spectrum of the Perron-Frobenius operator is degenerate for odd N.

Zeros of correlations in coupled map lattices are numerically studied.

Abstract

We show that, among all smooth one-dimensional maps conjugated to an N-ary shift (a Bernoulli shift of N symbols), Chebyshev maps are distinguished in the sense that they have least higher-order correlations. We generalise our consideration and study a family of shifted Chebyshev maps, presenting analytic results for two-point and higher-order correlation functions. We also review results for the eigenvalues and eigenfunctions of the Perron-Frobenius operator of N-th order Chebyshev maps and their shifted generalisations. The spectrum is degenerate for odd N. Finally, we consider coupled map lattices (CMLs) of shifted Chebyshev maps and numerically investigate zeros of the temporal and spatial nearest-neighbour correlations, which are of interest in chaotically quantized field theories.