Thomae's function and the space of ergodic measures

TL;DR

This paper investigates the structure of ergodic measures for a specific toral map, revealing that their organization resembles Thomae's function graph, thus connecting dynamical systems with a classical mathematical function.

Contribution

It demonstrates that the space of ergodic measures for the given map has a structure similar to Thomae's function graph, providing new insights into measure-theoretic dynamics.

Findings

The ergodic measures form a structure akin to Thomae's function graph.

The map studied is a skew translation on the torus.

The structure of ergodic measures is characterized in detail.

Abstract

We study the space of ergodic measures of the map $$f:\mathbb{T}^2\to \mathbb{T}^2, \ f(x, y)=(x, \ x+y)(\text{mod}\, 1),$$ and show that its structure is similar to the graph of Thomae's function.