Combinatorics. Measure-preserving dynamical system. In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Definition[edit] A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system with the following structure: is a set, is a σ-algebra over , is a probability measure, so that μ(X) = 1, and μ(∅) = 0, is a measurable transformation which preserves the measure , i.e., . , the identity function on X;, whenever all the terms are well-defined;, whenever all the terms are well-defined. The earlier, simpler case fits into this framework by definingTs = Ts for s ∈ N. The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem. Examples[edit] Examples include: Homomorphisms[edit] The concept of a homomorphism and an isomorphism may be defined.
Consider two dynamical systems and . The system is defined as. Maximal ergodic theorem. The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics. Suppose that is a probability space, that is a (possibly noninvertible) measure-preserving transformation, and that . Define by Then the maximal ergodic theorem states that for any λ ∈ R.
This theorem is used to prove the point-wise ergodic theorem. Keane, Michael; Petersen, Karl (2006), "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem", Institute of Mathematical Statistics Lecture Notes - Monograph Series, Institute of Mathematical Statistics Lecture Notes - Monograph Series 48: 248–251, doi:10.1214/074921706000000266, ISBN 0-940600-64-1 . Ergodic theory. Ergodic theory (Greek: έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics. The problem of metric classification of systems is another important part of the abstract ergodic theory. An outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of entropy for dynamical systems.
Ergodic transformations[edit] Ergodic theory is often concerned with ergodic transformations. Let T : X → X be a measure-preserving transformation on a measure space (X, Σ, μ), with μ(X) = 1. Then T is ergodic if for every E in Σ with T−1(E) = E, either μ(E) = 0 or μ(E) = 1. Examples[edit] Evolution of an ensemble of classical systems in phase space (top). Ergodic theorems[edit] Time average: This is defined as the average (if it exists) over iterations of T starting from some initial point x: where and. Ergodicity. In statistics, the term describes a random process for which the time average of one sequence of events is the same as the ensemble average.
In other words, for a Markov chain, as one increases the steps, there exists a positive probability measure at step that is independent of probability distribution at initial step 0 (Feller, 1971, p. 271).[1] Etymology[edit] The term "ergodic" was derived from the Greek words έργον (ergon: "work") and οδός (odos: "path" or "way").
It was chosen by Boltzmann while he was working on a problem in statistical mechanics.[2] Formal definition[edit] Let be a probability space, and be a measure-preserving transformation. (or alternatively that is ergodic with respect to T) if one of the following equivalent statements is true:[3] Measurable flows[edit] These definitions have natural analogues for the case of measurable flows and, more generally, measure-preserving semigroup actions. For each t ∈ R. Markov chains[edit] In a Markov chain, a state See also[edit]