Lebesgue Measure Preserving Thompson Monoid and Its Properties of Mixing, Periodic Points and Entropy

Abstract

This paper defines the Lebesgue measure preserving Thompson monoid, denoted by \(\mathbb{G}\), which is modeled on the Thompson group \(\mathbb{F}\) except that the elements of \(\mathbb{G}\) preserve the Lebesgue measure and can be non-invertible. The paper studies the properties of \(\mathbb{G}\) of mixing, periodic points, and entropy. Specifically, we first show that for any element of \(\mathbb{G}\) topological mixing (TM) is equivalent to locally eventually onto (LEO) and that the elements of \(\mathbb{G}\) whose elements are LEO are dense in the set of continuous measure preserving maps. Next, we show that every dyadic point is preperiodic and any map in \(\mathbb{G}\) is Markov. We show that for maps in a subset of \(\mathbb{G}\) there exist periodic points with period of 3, an essential feature of chaotic maps, and we further characterize periods of periodic points of other maps in \(\mathbb{G}\). Finally, we show that the elements of \(\mathbb{G}\) which are Markov LEO maps and whose entropy values are arbitrarily close to any number greater than or equal to 2 are dense in the set of continuous measure preserving maps.