Extensions of Laba‐Wang's condition for spectral pairs

Let ${\mu }_{M,D}$ be a self‐affine measure associated with an expanding matrix $M\in M_n\left(\mathbf{Z}\right)$ and a finite digit set $D?{\mathbf{Z}}^n$. We consider in this paper the spectrality of ${\mu }_{M,D}$. In the case when $(M^{?1}D,S)$ is a compatible pair for some $S?{\mathbf{Z}}^n$, a necessary condition is obtained for the spectral pair $({\mu }_{M,D},\Lambda (M,S))$. This condition is shown to be equivalent to the known necessary conditions for the same spectral pair. Moreover, we prove that all these necessary conditions are not sufficient in the higher dimensions, but they are sufficient in the dimension one. This extends Laba‐Wang's condition for spectral pairs.     

粗糙表面测量轮廓的分形插值模拟

引入粗糙表面的分形插值理论，提出用分形插值函数模拟粗糙表面轮廓的方法。并以车削，磨削及磨损表面为研究对象，对粗糙表面进行了分形插值模拟，系统地研究了影响其模拟模拟效果的因素和规律。发现可以采用分形插值理论来模拟或表征具有分形特征的粗糙表面。     

关于不变集的维数

本文讨论两类不变集的维数．在第一部分我们研究了平面上的一类自仿集,并给出了它的Hausdorff维数的—个估计．在第二部分我们研究Rd上的迭代函数系(IFS)的不变集, 特别我们考虑了压缩系数不是常数的情形,所得结果给出了经典结果的一个非平凡推广．     

Orthogonal exponentials on the generalized three-dimensional Sierpinski gasket

The self-affine measure μM,D corresponding to the expanding integer matrix

     

On new fractal phenomena connected with infinite linear IFS

We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. We show that the systems of cylinders of generalized Lüroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non‐faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary, we obtain the non‐faithfullness of the family of cylinders generated by the classical Lüroth expansion. We also develop new approach to the study of subsets of ‐essentially non‐normal numbers and prove that this set has full Hausdorff dimension. This result answers the open problem mentioned in 2 and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized Lüroth expansion.     

Markov chains in random environments and random iterated function systems

We consider random iterated function systems giving rise to Markov chains in random (stationary) environments. Conditions ensuring unique ergodicity and a ``pure type' characterization of the limiting ``randomly invariant' probability measure are provided. We also give a dimension formula and an algorithm for simulating exact samples from the limiting probability measure.

     

IFS中一个经典遍历性质的一些推广

该文针对概率迭代函数系统(IFS),给出一些遍历性质,这些结果推广了Elton[2]的结果,一个结果在某种意义上与Fustenberg[4]和Assani [1]关于弱混合系统中的结果类似.     

关于区间值直觉模糊集运算性质的注记

针对Atanassov[3]给出的区间值直觉模糊集运算,指出并修正一些区间值模糊集运算的错误性质,有利于以区间值直觉模糊集的进一步研究。     

IFS on boundaries of homogeneous trees: WQB condition and ultifractal analysis

We study, from the point of view of the multifractal analysis, iterated function systems on totally disconnected spaces, namely, the boundaries of homogeneous trees. In particular, we study in this setting the “weak quasi-Bernoulli” property introduced by Testud [3, 4]. After projection on ? or ?², we get new examples of self-similar measures which, being WQB, obey the multifractal formalism for positive q's.