Lebesgue Measure Preserving Thompson Monoid and Its Properties of Mixing, Periodic Points and Entropy | Chapter 03 | Innovations in Science and Technology Vol. 7

 This work defines the Lebesgue measure preserving Thompson monoid, designated by G, which is similar to the Thompson group F except that its members retain the Lebesgue measure and can be non-invertible. The work investigates the mixing qualities of G, periodic points, and entropy. To be more specific, we show that for each element of G, topological mixing (TM) is identical to locally eventually onto (LEO), and that the elements of G with LEO are dense in the set of continuous measure preserving mappings. Following that, we demonstrate that every dyadic point is preperiodic and that any map in G is Markov. We show that periodic points with period 3 exist for maps in a subset of G, which is a fundamental feature of chaotic maps, and we characterise the periods of periodic points in other maps in G. Finally, we prove that in the set of continuous measure preserving maps, the elements of G that are Markov LEO maps and whose entropy values are arbitrarily close to any number larger than or equal to 2 are dense.

Author(S) Details

William Li
Stanford University, Stanford, California 94305, US.

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