Image transformations, Markov operators, and sample median

TL;DR

This paper explores generalized Markov operators linked to iterated function systems, investigates their invariant measures, and introduces a generalized distribution of the sample median with invariance properties under various transformations.

Contribution

It extends classical fractal theory to broader Markov operators, defines a generalized sample median distribution, and establishes measure properties in low-dimensional spaces.

Findings

Unique invariant measures for contraction-based Markov operators on compact spaces.

Generalized distribution of sample median is equivariant under many transformations.

Topological measures on low-dimensional spaces are Radon measures.

Abstract

(I.) We consider generalizations of an iterated function system and the associated Markov operators. A Markov operator, defined on the space of (deficient) topological measures on a locally compact space, is an infinite convex linear combination of adjoints of (d-) image transformations. Restricted to measures, this Markov-Feller operator has a nonlinear dual operator given by an infinite convex linear combination of (conic) quasi-homomorphisms. If (d-) image transformations are contractions with respect to the Kantorovich-Rubinstein metric, a Markov operator has the unique invariant (deficient) topological measure. Taking a compact space, finitely many inverses of contractions as image transformations, and restricting the Markov operator to measures gives the classical result from the theory of fractals. There are various relations between Markov operator and the iterated function system where adjoints of (d-) image transformations are contractions on the compact metric space of $\{0,1\}$-valued (deficient) topological measures. For instance, the invariant (deficient) topological measure is the composition of the fixed point of the IFS and the basic (d-) image transformation. (II.) We define a generalized distribution of the sample median (g.d.s.m.) for continuous proper maps using an image transformation. We show that the g.d.s.m. and the inverse on the sample median are equivariant under solid variables, a large collection of transformations. On $\mathbb{R}^n$ such transformations include rotations, translations, symmetries, stretching, projections, monotone maps, etc. (III.) We show that a (signed) topological measure on a locally compact space with the covering dimension $\dim X \le 1$ is a (signed) Radon measure.