measure-preserving transformation
1Measure-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. Contents 1 Definition 2 Examples 3 Homomorphisms 4 …
2Lorentz transformation — A visualisation of the Lorentz transformation (full animation). Only one space coordinate is considered. The thin solid lines crossing at right angles depict the time and distance coordinates of an observer at rest with respect to that frame; the …
3Interval exchange transformation — In mathematics, an interval exchange transformation is a kind of dynamical system that generalises the idea of a circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several… …
4Ergodic theory — 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. A central concern of ergodic theory is the behavior of a dynamical …
5Dynamical system (definition) — This article presents the many ways to define a dynamical system. See the main article, dynamical system, for an overview of the topic. The dynamical system concept is a mathematical formalization for any fixed rule which describes the time… …
6Dynamical system — This article is about the general aspects of dynamical systems. For technical details, see Dynamical system (definition). For the study, see Dynamical systems theory. Dynamical redirects here. For other uses, see Dynamics (disambiguation). The… …
7Standard probability space — In probability theory, a standard probability space (called also Lebesgue Rokhlin probability space) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940 [1] . He showed that the unit interval endowed with… …
8Ergodicity — For other uses, see Ergodic (disambiguation). In mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. In physics the term is used to imply… …
9Poincaré recurrence theorem — In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence.… …
10Random dynamical system — In mathematics, a random dynamical system is a measure theoretic formulation of a dynamical system with an element of randomness , such as the dynamics of solutions to a stochastic differential equation. It consists of a base flow, the noise ,… …
