estimate of convolution transform of singular measure by approximate identity

Li, Cao; Lifeng, Xi; Zhang, Yiping

doi:10.1016/j.na.2013.08.001

Cited by 2 publications

(1 citation statement)

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“…Remark 1. The relation between the local dimension and P r (x − y) dµ(y) is discussed in a general framework by Wen and Zhang [7] and Cao et al [1]. Though Theorem 2 might follow from them, we give an independent proof so that our paper is self-contained.…”

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confidence: 89%

Spectral measure of the Thue–Morse sequence and the dynamical system and random walk related to it

Peng

Kamae

2015

Ergod. Th. Dynam. Sys.

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Let $1,-1,-1,1,-1,1,1,-1,-1,1,1,\ldots$ be the $\{-1,1\}$-valued Thue–Morse sequence. Its correlation dimension is $D_{2}$, satisfying $$\begin{eqnarray}\mathop{\sum }_{k=0}^{K-1}|{\it\gamma}(k)|^{2}\asymp K^{1-D_{2}}\end{eqnarray}$$ in the sense that the ratio between the left- and right-hand sides is bounded away from 0 and $\infty$ as $K\rightarrow \infty$, where ${\it\gamma}$ is the correlation function; its value is known [Zaks, Pikovsky and Kurths. On the correlation dimension of the spectral measure for the Thue–Morse sequence. J. Stat. Phys.88(5/6) (1997), 1387–1392] to be $$\begin{eqnarray}D_{2}=1-\log \frac{1+\sqrt{17}}{4}\bigg/\log 2=0.64298\ldots .\end{eqnarray}$$ Under its spectral measure ${\it\mu}$ on $[0,1)$, consider the transformation $T$ with $Tx=2x$ ($\text{mod}~1$). It is shown to be of Kolmogorov type having entropy at least $D_{2}\log 2$. Moreover, a random walk is defined by $T^{-1}$ which has the transition probability $$\begin{eqnarray}P_{1}((1/2)x+(1/2)j\mid x)=(1/2)(1-\cos ({\it\pi}(x+j)))\quad (j=0,1).\end{eqnarray}$$ It is proved that this random walk is mixing and ${\it\mu}$ is the unique stationary measure. Moreover, $$\begin{eqnarray}\lim _{N\rightarrow \infty }\int P_{N}((x-{\it\varepsilon},x+{\it\varepsilon})|x)\,d{\it\mu}(x)\asymp {\it\varepsilon}^{D_{2}}\quad (\text{as}~{\it\varepsilon}\rightarrow 0),\end{eqnarray}$$ where $P_{N}(\cdot \mid \cdot )$ is the $N$-step transition probability.

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confidence: 89%