Structure of the correlation function at the accumulation points of the logistic map

TL;DR

This paper investigates the detailed structure of the correlation function at the Feigenbaum point of the logistic map, supported by analytical and numerical methods, and extends the results to a broader class of attractors.

Contribution

It provides a rigorous analytical understanding of the correlation function's structure at the Feigenbaum point and generalizes the findings to other Feigenbaum attractors.

Findings

Correlation function structure is justified analytically.

Numerical experiments confirm the analytical results.

Generalization to arbitrary m·2^∞ Feigenbaum attractors.

Abstract

The correlation function of the trajectory exactly at the Feigenbaum point of the logistic map is investigated and checked by numerical experiments. Taking advantage of recent closed analytical results on the symbol-to-symbol correlation function of the generating partition, we are in position to justify the deep algorithmic structure of the correlation function apart from numerical constants. A generalization is given for arbitrary $m\cdot 2^{\infty}$ Feigenbaum attractors.