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arXiv · 2607.08676

Chaos in the Order of Finite Bernoulli Convolutions

Nadav Shrayer
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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.

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Nadav Shrayer. 2026-07-09. Chaos in the Order of Finite Bernoulli Convolutions. https://arxiv.org/abs/2607.08676

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