ERGODIC MAPPINGS WITH DISCRETE SPECTRUM

ABSTRACT

This Thesis is carried out on “Ergodic Mappings With Discrete Spectrum”. We will take our time to prove the main convergence theorems; Von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theorem. We shall look at Poincaré’s Recurrence Theorem, which comes into play when we discuss recurrence problems about the nature of orbits of points and measurable sets. Furthermore, we will discuss some necessary and sufficient conditions for an ergodic transformation T to have discrete spectrum. A physical quantity is said to have a discrete Spectrum if it only accepts distinct values with gaps between them. We shall discuss this concept of discrete spectrum, with ergodicity. We conclude by considering the work of Kushnirenko (1967) on "Metric Invariants Of Entropy Type", calculating the 2 n -sequence entropy of the two-dimensional Torus; (x, y) mod 1, T : R 2 → R 2 given by T(x, y) = (x+y, y), with the view to reprove his theorem in a shorter way, which is easier to study and conclude on isomorphisms of measure-preserving transformations and discrete spectrum.