Estimates on the dimension of self‐similar measures with overlaps

Abstract

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 ) $ \beta \in (1,2)$ , the dimension of the Bernoulli convolution μ β $ \mu _{\beta }$ satisfies dim ( μ β ) ⩾ 0.98040856 , \begin{equation*}\hskip7pc \dim (\mu _{\beta })\geqslant 0.98040856,\hskip-7pc\vspace*{-6pt} \end{equation*} which improves a previous uniform lower bound 0.82 obtained by Hare and Sidorov (Exp. Math. 27 (2018), no. 4, 414–418). This new uniform lower bound is very close to the known numerical approximation 0.98040931953 ± 10 − 11 $ 0.98040931953\pm 10^{-11}$ for dim μ β 3 $\dim \mu _{\beta _3}$ , where β 3 ≈ 1.839286755214161 $ \beta _{3} \approx 1.839286755214161$ is the largest root of the polynomial x 3 − x 2 − x − 1 $ x^{3}-x^{2}-x-1$ . Moreover, the infimum inf β ∈ ( 1 , 2 ) dim ( μ β ) $\inf _{\beta \in (1,2)}\dim (\mu _{\beta })$ is attained at a parameter β ∗ $\beta _*$ in a small interval ( β 3 − 10 − 8 , β 3 + 10 − 8 ) . \begin{equation*}\hskip7pc (\beta _{3} -10^{-8}, \beta _{3} + 10^{-8}).\hskip-7pc \end{equation*} When β $\beta$ is a Pisot number, we express dim ( μ β ) $\dim (\mu _{\beta })$ in terms of the measure-theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate dim ( μ β ) $\dim (\mu _\beta )$ from above as well.