1$ and $\mathcal{O}$ is an orthogonal matrix) and a finite subset $\mathcal{D}$ of $\mathbb{R}^n$, the pair $(A,\mathcal{D})$ determines a family of maps $\{f_d(x)=A^{-1}(x+d)\}_{d\in\mathcal{D}}$ which is referred to as a self-similar iterated function system (IFS), and it also determines a self-similar ..." />

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报告题目:

The visual representation of different types of measures for self-similar sets

报告人:

郑嘉(华中师范大学)

报告时间:

报告地点:

理学院东北楼四楼报告厅(404)

报告摘要:

For a similarity expanding matrix $A$ in $M_n(\mathbb{R})$ (i.e., $A = r \mathcal{O}$, where $|r|>1$ and $\mathcal{O}$ is an orthogonal matrix) and a finite subset $\mathcal{D}$ of $\mathbb{R}^n$, the pair $(A,\mathcal{D})$ determines a family of maps $\{f_d(x)=A^{-1}(x+d)\}_{d\in\mathcal{D}}$ which is referred to as a self-similar iterated function system (IFS), and it also determines a self-similar set $K$ which is the unique compact set satisfying the set-valued equation $AK=\bigcup_{d\in \mathcal{D}} (K+d)$. In this work, we define a measure

$$\mu=\lim_{N\rightarrow \infty}\sum_{d_0,d_1,\cdots d_{N-1}\in \mathcal{D}}\delta_{d_0+Ad_1+\cdots+A^{N-1}d_{N-1}}$$ associated with $(A,\mathcal{D})$ and establish a connection with the corresponding notions of upper and lower $s$-densities of $\mu$. This visual representation enables us to compute the precise value of several different type fractal measures such as packing measure, Hausdorff centered measure and spherical measure for self-similar sets, even if the IFS does not satisfy the OSC. Furthermore, based on Besicovitch's projection theorem, we construct a class of self-similar sets in $\mathbb{R}^n$ whose Hausdorff measure and packing measure are equal. This is a joint work with Xiaoye Fu and Hua Qiu.

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