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Concept# Measure-preserving dynamical system

Summary

In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, 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.
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 ,
is a probability measure, so that , and ,
is a measurable transformation which preserves the measure , i.e., .
One may ask why the measure preserving transformation is defined in terms of the inverse instead of the forward transformation . This can be understood in a fairly easy fashion. Consider a mapping of power sets:
Consider now the special case of maps which preserve intersections, unions and complements (so that it is a map of Borel sets) and also sends to (because we want it to be conservative). Every such conservative, Borel-preserving map can be specified by some surjective map by writing . Of course, one could also define , but this is not enough to specify all such possible maps . That is, conservative, Borel-preserving maps cannot, in general, be written in the form one might consider, for example, the map of the unit interval given by this is the Bernoulli map.
has the form of a pushforward, whereas is generically called a pullback. Almost all properties and behaviors of dynamical systems are defined in terms of the pushforward. For example, the transfer operator is defined in terms of the pushforward of the transformation map ; the measure can now be understood as an invariant measure; it is just the Frobenius–Perron eigenvector of the transfer operator (recall, the FP eigenvector is the largest eigenvector of a matrix; in this case it is the eigenvector which has the eigenvalue one: the invariant measure.

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