With these assumptions we have T defined for every integer n as a 1-1, onto, bimeasurable transformation. Henceforth we shall assume that every set considered is measurable, i.e. an element of a. We shall say that P is invariant if P(A) =P(TA) for every set A, P is ergodic if P is invariant and if P(U^L_oo TA) = 1 for every set A for which P(A) > 0 , and finally P is strongly mixing if P is inv...

Ergodic theory studies the long-term averaging properties of measurepreserving dynamical systems. In this paper, we state and present a proof of the ergodic theorem due to George Birkhoff, who observed the asymptotic equivalence of the time-average and space-average of a point x in a finite measure space. Then, we examine a number of applications of this theorem in number-theoretic problems, in...

This paper will explore the basics of discrete-time Markov chains used to prove the Ergodic Theorem. Definitions and basic theorems will allow us to prove the Ergodic Theorem without any prior knowledge of Markov chains, although some knowledge about Markov chains will allow the reader better insight about the intuitions behind the provided theorems. Even for those familiar with Markov chains, ...

In the thirty or so years since H. Furstenberg reproved Szemerédi’s theorem using methods from ergodic theory, many striking discoveries have been made in the area now known as Ergodic Ramsey theory. Perhaps the most surprising of these is the discovery that recurrence results can be obtained for polynomial sets, meaning sets of values of polynomials. The following pretty theorem, a special cas...

The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).

The ergodic hypothesis is a basic hypothesis typically invoked in atmospheric surface layer (ASL) experiments. The ergodic theorem of stationary random processes is introduced to analyse and verify the ergodicity of atmospheric turbulence measured using the eddy-covariance technique with two sets of field observational data. The results show that the ergodicity of atmospheric turbulence in atmo...

Contents Chapter 1. Introduction 5 Chapter 2. Measure-preserving transformations and other basic notions of ergodic theory for deterministic systems 11 1. Example: circle rotations 11 2. Measure-preserving transformations. Formal definitions 13 3. Poincaré's recurrence theorem 15 4. Stochastic processes: basic notions 17 5. Interpretation of measure-preserving maps via stationary processes 19 6...