Operators arising from invariant measures under some class of multidimensional transformations

TL;DR

This paper studies a linear operator linked to invariant measures under multidimensional transformations, providing explicit solutions and conditions for their existence, generalizing classical p-adic maps to higher dimensions.

Contribution

It introduces a new operator framework for invariant measures in higher-dimensional settings and derives explicit solutions and existence results.

Findings

Explicit solution formula for the functional equation

Existence of absolutely continuous invariant measures

Generalization of p-adic maps to higher dimensions

Abstract

We investigate a linear operator associated with a functional equation that arises from studying some class of invariant measures under multidimensional transformations. By examining its iterates, we derive an explicit solution formula for the functional equation in some class of functions and establish a result on the existence of an absolutely continuous invariant measure under a multidimensional transformation that can be viewed as a generalization of classical $p$-adic maps to higher dimensions.