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 $\beta$-expansions distribute in $\beta$, 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 $\beta$-expansions based on the lexicographical order of the $N$-binary words, and observe that for most sampled adjacent pairs when $\beta=2$, the distance in their order increases exponentially when $\beta$ 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 $\beta$-compactum with $\beta=g$ (g being the golden ratio) and binary reflected Gray code.