Papers › Estimates on the dimension of self-similar measures with overlaps
Estimates on the dimension of self-similar measures with overlaps
De-Jun Feng, Zhou Feng
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In this paper, we provide an algorithm to estimate from below the dimension of self-similar measures with overlaps. As an application, we show that for any β∈(1,2), the dimension of the Bernoulli convolution μᵦ satisfies (μᵦ) ≥0.9804085, which improves a previous uniform lower bound $0.82$ obtained by Hare and Sidorov \cite{HareSidorov2018}. This new uniform lower bound is very close to the known numerical approximation 0.98040931953±10⁻¹¹ for μ_(β₃), where β₃ ≈1.839286755214161 is the largest root of the polynomial x³-x²-x-1. Moreover, the infimum inf_(β∈(1,2))(μᵦ) is attained at a parameter β_* in a small interval (β₃ -10⁻⁸, β₃ + 10⁻⁸). When β is a Pisot number, we express (μᵦ) in terms of the measure-theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate (μᵦ) from above as well.
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