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.

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.

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.

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.

At the same time, these iterations will not compress or dilate any portion of the oatmeal: they preserve the measure that is density.

This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff.

Joining the first to the last claim and assuming that μ(X) is finite and nonzero, one has that for almost all x, i.e., for all x except for a set of measure zero.

is the trivial σ-algebra, and thus with probability 1: Von Neumann's mean ergodic theorem, holds in Hilbert spaces.

Then, for any x in H, we have: where the limit is with respect to the norm on H. In other words, the sequence of averages converges to P in the strong operator topology.

grows, while for the latter part, from the telescoping series one would have: This theorem specializes to the case in which the Hilbert space H consists of L2 functions on a measure space and U is an operator of the form where T is a measure-preserving endomorphism of X, thought of in applications as representing a time-step of a discrete dynamical system.

In fact, this result also extends to the case of strongly continuous one-parameter semigroup of contractive operators on a reflexive space.

If we pick a single complex number of unit length (which we think of as U), it is intuitive that its powers will fill up the circle.

The conditional expectation with respect to the sub-σ-algebra ΣT of the T-invariant sets is a linear projector ET of norm 1 of the Banach space Lp(X, Σ, μ) onto its closed subspace Lp(X, ΣT, μ).

Finally, if ƒ is assumed to be in the Zygmund class, that is |ƒ| log+(|ƒ|) is integrable, then the ergodic means are even dominated in L1.

Another consequence of the ergodic theorem is that the average recurrence time of A is inversely proportional to the measure of A, assuming[clarification needed] that the initial point x is in A, so that k0 = 0.

The ergodicity of the geodesic flow on compact Riemann surfaces of variable negative curvature and on compact manifolds of constant negative curvature of any dimension was proved by Eberhard Hopf in 1939, although special cases had been studied earlier: see for example, Hadamard's billiards (1898) and Artin billiard (1924).

The relation between geodesic flows on Riemann surfaces and one-parameter subgroups on SL(2, R) was described in 1952 by S. V. Fomin and I. M. Gelfand.

Ergodicity of the geodesic flow on Riemannian symmetric spaces was demonstrated by F. I. Mautner in 1957.

G. Sinai proved ergodicity of the geodesic flow on compact manifolds of variable negative sectional curvature.

In the 1930s G. A. Hedlund proved that the horocycle flow on a compact hyperbolic surface is minimal and ergodic.