Ergodic theory

A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set. More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important theorems are those of Birkhoff (1931) and von Neumann which assert the existence of a time average along each trajectory. For the special class of ergodic systems, this time average is the same for almost all initial points: statistically speaking, the system that evolves for a long time 'forgets' its initial state. Stronger properties, such as mixing and equidistribution, have also been extensively studied. 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. The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory. The underlying idea is that for certain systems the time average of their properties is equal to the average over the entire space. Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind. In geometry, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context for applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the theory of diophantine approximations, L-functions). Ergodic theory is often concerned with ergodic transformations. The intuition behind such transformations, which act on a given set, is that they do a thorough job 'stirring' the elements of that set (e.g., if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not allow the syrup to remain in a local subregion of the oatmeal, but will distribute the syrup evenly throughout. At the same time, these iterations will not compress or dilate any portion of the oatmeal: they preserve the measure that is density.) Here is the formal definition. 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. Let T: X → X be a measure-preserving transformation on a measure space (X, Σ, μ) and suppose ƒ is a μ-integrable function, i.e. ƒ ∈ L1(μ). Then we define the following averages: In general the time average and space average may be different. But if the transformation is ergodic, and the measure is invariant, then the time average is equal to the space average almost everywhere. This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff. (Actually, Birkhoff's paper considers not the abstract general case but only the case of dynamical systems arising from differential equations on a smooth manifold.) The equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval. More precisely, the pointwise or strong ergodic theorem states that the limit in the definition of the time average of ƒ exists for almost every x and that the (almost everywhere defined) limit function ƒ̂ is integrable: