Symbol-to-symbol correlation function at the Feigenbaum point of the logistic map

TL;DR

This paper introduces and rigorously analyzes the symbol-to-symbol correlation function at the Feigenbaum point of the logistic map, combining analytical predictions with numerical verification to understand correlations in multifractal attractors.

Contribution

It provides the first rigorous definition and analytical calculation of the symbol-to-symbol correlation function for the Feigenbaum attractor, validated by numerical experiments.

Findings

Analytical predictions match numerical results

Revealed the complex structure of the correlation function

Connected correlation decay to multifractal properties

Abstract

Recently, simple dynamical systems such as the 1-d maps on the interval, gained significant attention in the context of statistical physics and complex systems. The decay of correlations in these systems, can be characterized and measured by correlation functions. In the context of symbolic dynamics of the non-chaotic multifractal attractors (i.e. Feigenbaum attractors), one observable, the symbol-to-symbol correlation function, for the generating partition of the logistic map, is rigorously introduced and checked with numerical experiments. Thanks to the Metropolis-Stein-Stein (MSS) algorithm this observable can be calculated analytically, giving predictions in absolute accordance with numerical computations. The deep, algorithmic structure of the observable is revealed clearly reflecting the complexity of the multifractal attractor.