Estimates on the dimension of self‐similar measures with overlaps
Estimates on the dimension of self‐similar measures with overlaps
复制标题
具有重叠的自相似测度维度的估计
DOI：
10.1112/jlms.12555
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Zhou Feng
中科院分区：
文献类型：
--
作者：
De;Zhou Feng
在本文中，我们提供了一种算法来从下面估计具有重叠的自相似度量的维度。作为一个应用，我们证明对于任何 βε(1,2)$ \beta \in (1,2)$ ，伯努利卷积的维数 μβ$ \mu _{\beta }$ 满足 dim(μβ)⩾0.98040856,\begin{equation*}\hskip7pc \dim (\mu _{\beta })\geqslant 0.98040856,\hskip-7pc\vspace*{-6pt} \end{equation*}改进了 Hare 和 Sidorov 先前获得的统一下界 0.82 (Exp. Math. 27 (2018), no. 4, 414–418)。这个新的统一下界非常接近已知的数值近似 0.98040931953±10−11$ 0.98040931953\pm 10^{-11}$ 对于 dimμβ3$\dim \mu _{\beta _3}$ ，其中 β3≈1.839286755214161$ \beta _{3} \approx 1.839286755214161$ 是多项式 x3−x2−x−1$ x^{3}-x^{2}-x-1$ 的最大根。此外，下确界 infβε(1,2)dim(μβ)$\inf _{\beta \in (1,2)}\dim (\mu _{\beta })$ 在小区间 (β3−10−8,β3+10−8) 内的参数 β*$\beta _*$ 处获得。\begin{equation*}\hskip7pc (\beta _{3} -10^{-8}, \beta _{3} + 10^{-8}).\hskip-7pc \end{equation*}当 β$\beta$ 为皮索数时，我们用某个矩阵压力函数的平衡测度的测度理论熵来表示 dim(μβ)$\dim (\mu _{\beta })$，并提出了一种估计 dim(μβ)$\dim (\mu _\beta )$ 也来自上面。
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 μβ$ \mu _{\beta }$ satisfies dim(μβ)⩾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 dimμβ3$\dim \mu _{\beta _3}$ , where β3≈1.839286755214161$ \beta _{3} \approx 1.839286755214161$ is the largest root of the polynomial x3−x2−x−1$ x^{3}-x^{2}-x-1$ . Moreover, the infimum infβ∈(1,2)dim(μβ)$\inf _{\beta \in (1,2)}\dim (\mu _{\beta })$ is attained at a parameter β∗$\beta _*$ in a small interval (β3−10−8,β3+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(μβ)$\dim (\mu _{\beta })$ in terms of the measure‐theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate dim(μβ)$\dim (\mu _\beta )$ from above as well.
影响因子：
1.7
作者：
Kleptsyn V
通讯作者：
Kleptsyn V