Renormalization group structure for sums of variables generated by incipiently chaotic maps

TL;DR

This paper uncovers a renormalization group framework for the distribution of sums of chaotic variables in unimodal maps, revealing a universal multifractal fixed point and explaining the transition to Gaussian behavior.

Contribution

It introduces a novel RG structure for sums of deterministic chaotic variables, identifying a universal multifractal fixed point related to the Feigenbaum attractor.

Findings

Identifies a nontrivial multifractal fixed point in the distribution of sums.

Explains the crossover from multifractal to Gaussian distributions.

Connects the flow to chaotic band merging sequences.

Abstract

We look at the limit distributions of sums of deterministic chaotic variables in unimodal maps and find a remarkable renormalization group (RG) structure associated to the operation of increment of summands and rescaling. In this structure - where the only relevant variable is the difference in control parameter from its value at the transition to chaos - the trivial fixed point is the Gaussian distribution and a novel nontrivial fixed point is a multifractal distribution that emulates the Feigenbaum attractor, and is universal in the sense of the latter. The crossover between the two fixed points is explained and the flow toward the trivial fixed point is seen to be comparable to the chaotic band merging sequence. We discuss the nature of the Central Limit Theorem for deterministic variables.