Rank as a function of measure

TL;DR

This paper investigates the topological properties of the rank function on invariant measures in dynamical systems, showing it is of Young class LU, meaning it can be approximated by increasing sequences of upper semicontinuous functions.

Contribution

It establishes that the rank function is of Young class LU, providing new insights into its topological structure in dynamical systems.

Findings

Rank is of Young class LU.

Rank can be approximated by increasing sequences of upper semicontinuous functions.

Topological properties of the rank function are characterized.

Abstract

We establish certain topological properties of rank understood as a function on the set of invariant measures on a topological dynamical system. To be exact, we show that rank is of Young class LU (i.e., it is the limit of an increasing sequence of upper semicontinuous functions)