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

This paper defines the Lebesgue measure preserving Thompson monoid, denoted by , which is modeled on the Thompson group except that the elements of preserve the Lebesgue measure and can be non-invertible. The paper studies the properties of of mixing, periodic points, and entropy. Specifically, we first show that for any element of topological mixing (TM) is equivalent to locally eventually onto (LEO) and that the elements of 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 is Markov. We show that for maps in a subset of 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 . Finally, we show that the elements of 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.