Invariant Probability Measures under $p$-adic Transformations

TL;DR

This paper characterizes all invariant probability measures under $p$-adic transformations, including atomic, continuous, and singular measures, using iterative functional equations to describe their structure.

Contribution

It provides a comprehensive description of all invariant measures under $p$-adic transformations, extending beyond the Lebesgue measure to include atomic and singular measures.

Findings

Identified all atomic invariant measures under $p$-adic transformations.

Described continuous and singular invariant measures.

Used iterative functional equations to analyze measure structures.

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.