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, ...

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).

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...

Let S = {ki,ki, ...} be an increasing sequence of positive integers. We call S strongly ergodic if for every measure preserving transformation T on a probability space (Cl, J, P) and every / £ Li(f2) we have limn-»oo(l/n) J^^j f(TkiuJ) = Pf(w) a.e. where Pf is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence S to be strongly ergodi...

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...

The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for σ-finite invariant measures (Corollary 1). For infinite measures the ergodic decomposition is not unique, but the measure class of the decomposing measure on the space of projective measures is uniquely defined by ...

Let T be a positive L1-L∞ contraction. We prove that the following statements are equivalent in constructive mathematics. 1. The projection in L2 on the space N : = cl{x−T x: x∈L2} exists; 2. The sequence (T n)n∈N Cesàro-converges in the L2 norm; 3. The sequence (T n)n∈N Cesàro-converges almost everywhere. Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Du...

Gases reach equilibrium when left to themselves. Why do they behave in this way? The canonical answer to this question, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer has been criticised on different grounds and is now widely regarded as flawed. In this paper we argue that some of the main arguments against Boltzmann’s answer, in particular, arguments bas...

Above is the famous Fekete’s lemma which demonstrates that the ratio of subadditive sequence (an) to n tends to a limit as n approaches infinity. This lemma is quite crucial in the field of subadditive ergodic theorems because it gives mathematicians some general ideas and guidelines in the non-random setting and leads to analogous discovery in the random setting. Kingman’s Subadditive Ergodic ...