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arXiv · 2412.06406

Invariant Probability Measures under $p$-adic Transformations

Oleksandr V. Maslyuchenko·Janusz Morawiec·Thomas Zürcher
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Abstract

It is well-known that the Lebesgue measure is the unique absolutely continuous invariant probability measure under the $p$-adic transformation. The purpose of this paper is to characterize the family of all invariant probability measures under the $p$-adic transformation and to provide some description of them. In particular, we describe the subfamily of all atomic invariant measures under the $p$-adic transformation as well as the subfamily of all continuous and singular invariant probability measures under the $p$-adic transformation. Iterative functional equations play the base role in our considerations.

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Oleksandr V. Maslyuchenko, Janusz Morawiec, Thomas Zürcher. 2024-12-09. Invariant Probability Measures under $p$-adic Transformations. https://arxiv.org/abs/2412.06406

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