Invariant Probability Measures under $p$-adic Transformations
Classical Analysis and ODEs
2025-06-03 v2
Abstract
It is well-known that the Lebesgue measure is the unique absolutely continuous invariant probability measure under the -adic transformation. The purpose of this paper is to characterize the family of all invariant probability measures under the -adic transformation and to provide some description of them. In particular, we describe the subfamily of all atomic invariant measures under the -adic transformation as well as the subfamily of all continuous and singular invariant probability measures under the -adic transformation. Iterative functional equations play the base role in our considerations.
Keywords
Cite
@article{arxiv.2412.06406,
title = {Invariant Probability Measures under $p$-adic Transformations},
author = {Oleksandr V. Maslyuchenko and Janusz Morawiec and Thomas Zürcher},
journal= {arXiv preprint arXiv:2412.06406},
year = {2025}
}