Uniform lower bounds on the dimension of Bernoulli convolutions

2021/02/15 by Kleptsyn, Victor, Pollicott, Mark, Vytnova, Polina
#11K55 #37F35 #37M25 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2102.07714

Abstract

In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure μ_λ is the probability measure corresponding to the law of the random variable ξ= ∑_k=0^∞ ξ_kλ^k, where ξ_k are i.i.d. random variables assuming values -1 and 1 with equal probability and \frac12 < λ< 1. In particular, for Bernoulli convolutions we give a uniform lower bound dim_H(μ_λ) ≥ 0.96399 for all \frac12

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