Lebesgue Measure Preserving Thompson's Monoid

TL;DR

This paper introduces a new monoid of Lebesgue measure preserving, non-invertible transformations modeled on Thompson's group, and explores its algebraic and dynamical properties.

Contribution

It defines the Lebesgue measure preserving Thompson's monoid and analyzes its unique algebraic and dynamical characteristics compared to Thompson's group.

Findings

The monoid exhibits distinct algebraic properties from Thompson's group.

It demonstrates various dynamical behaviors including mixing and periodicity.

The paper characterizes generators and topological conjugacy within the monoid.

Abstract

This paper defines Lebesgue measure preserving Thompson's monoid, denoted by $\mathbb{G}$, which is modeled on Thompson's group $\mathbb{F}$ except that the elements of $\mathbb{G}$ are non-invertible. Moreover, it is required that the elements of $\mathbb{G}$ preserve Lebesgue measure. Monoid $\mathbb{G}$ exhibits very different properties from Thompson's group $\mathbb{F}$. The paper studies a number of algebraic (group-theoretic) and dynamical properties of $\mathbb{G}$ including approximation, mixing, periodicity, entropy, decomposition, generators, and topological conjugacy.