“分形几何及应用” 学术报告会

$$\mu_{\{A_n, n\ge 1\}}:=\delta_{A_1^{-1}\D}\ast\delta_{A_1^{-1}A_2^{-2}\D}\ast\cdots$$

is a probability measure with compact support, where $\D=\{0, (1, 0)^t, (0, 1)^t\}$ is the Sierpinski digit. We prove that there exists a set $\Lambda\subset\R^2$ such that the family $\{e^{2\pi i <\lambda,x>}: \lambda\in\Lambda\}$ is an orthonormal basis of $L^2( \mu_{\{A_n, n\ge 1\}})$ if and only if $\frac 13(1, -1)A_n\in\Z^2$ for $n\ge 2$ under some metric conditions on $A_n$.

with $0\in\mathcal{D}$. The family of maps $\{f_d(x)=A^{-1}(x+d)\}_{d\in\mathcal{D}}$ is called a self-affine iterated function system (self-affine IFS). The self-affine set $K=K(A,\mathcal{D})$ is the unique compact set determined by $(A, {\mathcal D})$ satisfying the set-valued equation $K=\displaystyle\bigcup_{d\in\mathcal{D}}f_d(K)$.

The number $s=n\,\ln(\# \mathcal{D})/\ln(q)$ with $q=|\det(A)|$, is the so-called pseudo similarity dimension of $K$. As shown by He and Lau, one can associate with $A$ and any number $s\ge 0$ a natural pseudo Hausdorff measure denoted by $\mathcal{H}_w^s.$ In this paper, we show that, if $s$ is chosen to be the pseudo similarity dimension of $K$, then the condition $\mathcal{H}_w^s(K)> 0$ holds if and only if the IFS $\{f_d\}_{d\in\mathcal{D}}$ satisfies the open set condition (OSC). This extends the well-known result for the self-similar case that the OSC is equivalent to $K$ having positive Hausdorff measure $\mathcal{H}^s$ for a suitable $s$.

Furthermore, we relate the exact value of pseudo Hausdorff measure $\mathcal{H}_w^s(K)$ to a notion of upper $s$-density with respect to the pseudo norm $w(x)$ associated with $A$ for the measure

$\mu=\lim\limits_{M\to\infty}\sum\limits_{d_0,\dotsc,d_{M-1}\in\mathcal{D}}\delta_{d_0 + Ad_1 + \dotsb + A^{M-1}d_{M-1}}$ in the case that $\#\mathcal{D}\le\lvert\det A\rvert$.

In this talk, we will consider the spectrality of generalized Sierpinski-type self-affine measures, and give the necessary and sufficient conditions for it to be a spectral measure.