Circularly invariant uniformizable probability measures for linear transformations

TL;DR

This paper establishes a threshold condition for the existence and uniqueness of circularly invariant uniformizable probability measures under linear transformations with non-zero slope, based on the support's diameter.

Contribution

It proves a threshold criterion for the existence and uniqueness of CIUPMs for linear transformations, depending solely on the support diameter.

Findings

Existence of CIUPMs depends on support diameter exceeding a slope-dependent threshold.

CIUPMs are unique up to translation when support diameter equals the threshold.

The threshold constant c is determined solely by the slope of the linear transformation.

Abstract

In this paper, we prove a threshold result on the existence of a circularly invariant uniformizable probability measure (CIUPM) for linear transformations with non-zero slope on the line. We show that there is a threshold constant $c$ depending only on the slope of the linear transformation such that there exists a CIUPM if and only if its support has a diameter at least as large as $c.$ Moreover, the CIUPM is unique up to translation if the diameter of the support equals $c.$