Continuous $\times p,\times q$-invariant measures on the unit circle

TL;DR

This paper characterizes continuous measures on the unit circle invariant under multiplication by p and q, showing they can be approximated by irrational orbits and described by specific functional equations.

Contribution

It provides a new representation of continuous  p, q-invariant measures using simple forms and functional equations, linking orbit averages and measure properties.

Findings

Invariant measures are limits of averages along irrational orbits

Such measures correspond to solutions of specific functional equations

The work bridges orbit dynamics and measure theory on the circle

Abstract

We express continuous $\times p,\times q$-invariant measures on the unit circle via some simple forms. On one hand, a continuous $\times p,\times q$-invariant measure is the weak-$*$ limit of average of Dirac measures along an irrational orbit. On the other hand, a continuous $\times p,\times q$-invariant measure is a continuous function on $[0,1]$ satisfying certain function equations.