Chaos in the Order of Finite Bernoulli Convolutions

Abstract

In this note we explore numerically the finite Bernoulli convolutions. We show that with a suitable choice of parameter, it might serve as a toy model for intermittent energy cascade in fully developed turbulence. We then show how the crossings of $β$-expansions distribute in $β$, and suggest that it might highlight the parameters with enhanced overlap structure that are related to measures that are singular continuous. We later introduce a notion of order to the $β$-expansions based on the lexicographical order of the $N$-binary words, and observe that for most sampled adjacent pairs when $β=2$, the distance in their order increases exponentially when $β$ decreases from 2 to 1. This suggests 'chaotic' behavior, with the 'Lyapunov exponents' bunched into several clusters that depend on $N$. We end the note with some 'order plots' and an interesting connection between the finite $β$-compactum with $β=g$ (g being the golden ratio) and binary reflected Gray code.