Ostrogradsky-Sierpi\'nski-Pierce expansion: dynamical systems, probability theory and fractal geometry points of view

Abstract

We establish several new probabilistic, dynamical, dimensional and number theoretical phenomena connected with Ostrogradsky-Sierpi\'nski-Pierce expansion. First of all, we develop metric, ergodic and dimensional theories of the Ostrogradsky-Sierpi\'nski-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-Sierpi\'nski-Pierce expansion. Properties of the symbolic dynamical system generated by a shift-transformation $T$ on the difference-version of the Ostrogradsky-Sierpi\'nski-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 $\eta$ with independent identically distributed differences of the Ostrogradsky-Sierpi\'nski-Pierce expansion. Necessary and sufficient conditions for $\eta$ to be discrete resp. singularly continuous are found. We prove that $\eta$ can not be absolutely continuously distributed.