1. Invariant Probability Measures under $p$-adic Transformations
- Author
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Morawiec, Janusz and Zürcher, Thomas
- Subjects
Mathematics - Classical Analysis and ODEs ,Primary 37E05, 39B12, Secondary 26A30, 28D05 - 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.
- Published
- 2024