In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a measure-preserving dynamical system is an object of study in the abstract formulation of
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
, and
ergodic theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
in particular. Measure-preserving systems obey the
Poincaré recurrence theorem
In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for ...
, and are a special case of
conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from
classical mechanics (in particular, most
non-dissipative systems) as well as systems in
thermodynamic equilibrium.
Definition
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
,
*