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Operators arising from invariant measures under some class of multidimensional transformations

Oleksandr V. Maslyuchenko, Janusz Morawiec, Thomas Zürcher

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.

Operators arising from invariant measures under some class of multidimensional transformations

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 -adic maps to higher dimensions.
Paper Structure (11 sections, 9 theorems, 114 equations)

This paper contains 11 sections, 9 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multidimensional increment of functions
  4. Gradient with respect to groups of variables
  5. Multidimensional analogue of Lipschitz functions
  6. The gradient increment and the average gradient of multidimensional Lipschitz functions
  7. Multidimensional MW-operators
  8. Iterations of the MW-operators
  9. The case of affine functions
  10. Multivariate distribution of a Borel measure
  11. Invariant measures under some class of piecewise affine functions

Key Result

Proposition 3.1

Assume that $m\in N$ and let $f\colon D\to\mathbb{R}$. Then for all $x=(x_n)_{n\in N}\in D$ and $y=(y_n)_{n\in N}\in D$ such that $\Pi(x,y)\subseteq D$.

Theorems & Definitions (25)

  • Proposition 3.1
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • Definition 7.1: admissible
  • Definition 7.2: multidimensional MW-operator
  • Definition 7.3: multidimensional MW-functional equation
  • Example 7.4
  • ...and 15 more