Stationary distributions of sums of marginally chaotic variables as renormalization group fixed points

TL;DR

This paper introduces a renormalization group approach to analyze the limit distributions of sums of chaotic variables in unimodal maps, revealing fixed points corresponding to Gaussian and multifractal distributions.

Contribution

It develops a novel RG framework that links the control parameter near chaos to the distribution of sums, identifying fixed points and crossover behavior.

Findings

Gaussian as trivial fixed point

Multifractal distribution as nontrivial fixed point

Sequence of chaotic band mergers leading to Gaussian distribution

Abstract

We determine the limit distributions of sums of deterministic chaotic variables in unimodal maps assisted by a novel renormalization group (RG) framework associated to the operation of increment of summands and rescaling. In this framework the difference in control parameter from its value at the transition to chaos is the only relevant variable, the trivial fixed point is the Gaussian distribution and a nontrivial fixed point is a multifractal distribution with features similar to those of the Feigenbaum attractor. The crossover between the two fixed points is discussed and the flow toward the trivial fixed point is seen to consist of a sequence of chaotic band mergers.