Ostrogradsky-Sierpierski-Pierce展开：动力系统，概率论和分形几何观点
Ostrogradsky-Sierpiński-Pierce expansion: dynamical systems, probability theory and fractal geometry points of view
摘要：
我们建立了几个新的概率，动力学，维数和数论现象与Ostrogradsky-Sierpienski-Pierce扩展。首先，我们发展了Ostrogradsky-Sierpierski-Pierce展开的度量、遍历和维数理论。特别是，它被证明，几乎所有的勒贝格实数的任何数字$i$从字母表$A= \mathbb{N} $只出现在Ostrogradsky-Sierpienski-Pierce展开的差异版本的许多次。本文还详细研究了由Ostrogradsky-Sierpierski-Pierce展开的差分形式上的移位变换$T$所生成的符号动力系统的性质。证明了不存在不变和遍历的概率测度（w.r.t. $T$）和绝对连续（w.r.t.勒贝格测度）。第三，研究了具有Ostrogradsky-Sierpierski-Pierce展开式的独立同分布差的随机变量$η$的性质。分别给出了$η$离散的充要条件和离散的充要条件。证明了$η$不可能是绝对连续分布的.
英文摘要：
We establish several new probabilistic, dynamical, dimensional and number theoretical phenomena connected with Ostrogradsky-Sierpiński-Pierce expansion. First of all, we develop metric, ergodic and dimensional theories of the Ostrogradsky-Sierpiński-Pierce expansion. In particular, it is proven that for Lebesgue almost all real numbers any digit $i$ from the alphabet $A= \mathbb{N} $ appears only finitely many times in the difference-version of the Ostrogradsky-Sierpiński-Pierce expansion. Properties of the symbolic dynamical system generated by a shift-transformation $T$ on the difference-version of the Ostrogradsky-Sierpiński-Pierce expansion are also studied in details. It is shown that there are no probability measures which are invariant and ergodic (w.r.t. $T$) and absolutely continuous (w.r.t. Lebesgue measure). Thirdly, we study properties of random variables $η$ with independent identically distributed differences of the Ostrogradsky-Sierpiński-Pierce expansion. Necessary and sufficient conditions for $η$ to be discrete resp. singularly continuous are found. We prove that $η$ can not be absolutely continuously distributed.
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