Ergodisk

Ergodisk is a coined term used in discussions of ergodic theory to describe a disk-shaped dynamical system that exhibits ergodic behavior. In this usage, an ergodisk consists of a closed unit disk D in the plane equipped with a probability measure μ on its Borel sets and a μ-measurable map T: D → D that preserves μ.

Such a triple (D, μ, T) is called an ergodisk when T is ergodic with respect to μ, meaning that every measurable set E with T^{-1}(E) = E has μ(E) equal to 0 or 1. Equivalently, time averages of integrable functions converge μ-almost everywhere to space averages, as guaranteed by Birkhoff's ergodic theorem for ergodic systems.

Concrete constructions of an ergodisk typically involve maps that mix the angular and radial coordinates in a nontrivial way, ensuring μ is preserved and invariant sets are trivial. While the term ergodisk is not standard in the literature, such models are used pedagogically to illustrate how ergodic behavior can arise on a simple, bounded domain.

Relation to other concepts: an ergodisk is a specific instance of a measure-preserving dynamical system on a compact metric space. It connects to broader topics in ergodic theory, such as equidistribution, mixing, skew products, and the study of invariant measures.

See also: ergodic theory; measure-preserving transformation; ergodicity; Birkhoff's theorem; dynamical system.

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