Measure-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. 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.
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:
Discussion.
One may ask why the measure preserving transformation is defined in terms of the inverse formula_11 instead of the forward transformation formula_12. This can be understood in a fairly easy fashion. Consider a mapping formula_13 of power sets:
Consider now the special case of maps formula_13 which preserve intersections, unions and complements (so that it is a map of Borel sets) and also sends formula_2 to formula_2 (because we want it to be conservative). Every such conservative, Borel-preserving map can be specified by some surjective map formula_18 by writing formula_19. Of course, one could also define formula_20, but this is not enough to specify all such possible maps formula_13. That is, conservative, Borel-preserving maps formula_13 cannot, in general, be written in the form formula_23 Obviously! one might say; consider, for example, the map of the unit interval formula_24 given by formula_25 this is the Bernoulli map.
Note that formula_26 has the form of a pushforward, whereas formula_27 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 formula_28; the measure formula_9 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.)
There are two classification problems of interest. One, discussed below, fixes formula_30 and asks about the isomorphism classes of a transformation map formula_28. The other, discussed in transfer operator, fixes formula_32 and formula_28, and asks about maps formula_9 that are measure-like. Measure-like, in that they preserve the Borel properties, but are no longer invariant; they are in general dissipative and so give insights into dissipative systems and the route to equilibrium. 
In terms of physics, the measure-preserving dynamical system formula_1 often describes a physical system that is in equilibrium, for example, thermodynamic equilibrium. One might ask: how did it get that way? Often, the answer is by stirring, mixing, turbulence, thermalization or other such processes. If a transformation map formula_28 describes this stirring, mixing, etc. then the system formula_1 is all that is left, after all of the transient modes have decayed away. The transient modes are precisely those eigenvectors of the transfer operator that have eigenvalue less than one; the invariant measure formula_9 is the one mode that does not decay away. The rate of decay of the transient modes are given by (the logarithm of) their eigenvalues; the eigenvalue one corresponds to infinite half-life.
Informal example.
The microcanonical ensemble from physics provides an informal example. Consider, for example, a fluid, gas or plasma in a box of width, length and height formula_39 consisting of formula_40 atoms. A single atom in that box might be anywhere, having arbitrary velocity; it would be represented by a single point in formula_41 A given collection of formula_40 atoms would then be a "single point" somewhere in the space formula_43 The "ensemble" is the collection of all such points, that is, the collection of all such possible boxes (of which there are an uncountably-infinite number). This ensemble of all-possible-boxes is the space formula_2 above. 
In the case of an ideal gas, the measure formula_9 is given by the Maxwell–Boltzmann distribution. It is a product measure, in that if formula_46 is the probability of atom formula_47 having position and velocity formula_48, then, for formula_40 atoms, the probability is the product of formula_40 of these. This measure is understood to apply to the ensemble. So, for example, one of the possible boxes in the ensemble has all of the atoms on one side of the box. One can compute the likelihood of this, in the Maxwell–Boltzmann measure. It will be enormously tiny, of order formula_51 Of all possible boxes in the ensemble, this is a ridiculously small fraction.
The only reason that this is an "informal example" is because writing down the transition function formula_28 is difficult, and, even if written down, it is hard to perform practical computations with it. Difficulties are compounded if the interaction is not an ideal-gas billiard-ball type interaction, but is instead a van der Waals interaction, or some other interaction suitable for a liquid or a plasma; in such cases, the invariant measure is no longer the Maxwell–Boltzmann distribution. The art of physics is finding reasonable approximations.
This system does exhibit one key idea from the classification of measure-preserving dynamical systems: two ensembles, having different temperatures, are inequivalent. The entropy for a given canonical ensemble depends on its temperature; as physical systems, it is "obvious" that when the temperatures differ, so do the systems. This holds in general: systems with different entropy are not isomorphic.
Examples.
 Unlike the informal example above, the examples below are sufficiently well-defined and tractable that explicit, formal computations can be performed. 
Generalization to groups and monoids.
The definition of a measure-preserving dynamical system can be generalized to the case in which "T" is not a single transformation that is iterated to give the dynamics of the system, but instead is a monoid (or even a group, in which case we have the action of a group upon the given probability space) of transformations "Ts" : "X" → "X" parametrized by "s" ∈ Z (or R, or N ∪ {0}, or [0, +∞)), where each transformation "Ts" satisfies the same requirements as "T" above. In particular, the transformations obey the rules:
The earlier, simpler case fits into this framework by defining "Ts" = "Ts" for "s" ∈ N.
Homomorphisms.
The concept of a homomorphism and an isomorphism may be defined.
Consider two dynamical systems formula_56 and formula_57. Then a mapping
is a homomorphism of dynamical systems if it satisfies the following three properties:
The system formula_57 is then called a factor of formula_56.
The map formula_67 is an isomorphism of dynamical systems if, in addition, there exists another mapping
that is also a homomorphism, which satisfies
Hence, one may form a category of dynamical systems and their homomorphisms.
Generic points.
A point "x" ∈ "X" is called a generic point if the orbit of the point is distributed uniformly according to the measure.
Symbolic names and generators.
Consider a dynamical system formula_75, and let "Q" = {"Q"1, ..., "Qk"} be a partition of "X" into "k" measurable pair-wise disjoint pieces. Given a point "x" ∈ "X", clearly "x" belongs to only one of the "Qi". Similarly, the iterated point "Tnx" can belong to only one of the parts as well. The symbolic name of "x", with regards to the partition "Q", is the sequence of integers {"a""n"} such that
The set of symbolic names with respect to a partition is called the symbolic dynamics of the dynamical system. A partition "Q" is called a generator or generating partition if μ-almost every point "x" has a unique symbolic name.
Operations on partitions.
Given a partition Q = {"Q"1, ..., "Q""k"} and a dynamical system formula_75, define the "T"-pullback of "Q" as
Further, given two partitions "Q" = {"Q"1, ..., "Qk"} and "R" = {"R"1, ..., "R""m"}, define their refinement as
With these two constructs, the "refinement of an iterated pullback" is defined as
which plays crucial role in the construction of the measure-theoretic entropy of a dynamical system.
Measure-theoretic entropy.
The entropy of a partition formula_81 is defined as
The measure-theoretic entropy of a dynamical system formula_75 with respect to a partition "Q" = {"Q"1, ..., "Q""k"} is then defined as
Finally, the Kolmogorov–Sinai metric or measure-theoretic entropy of a dynamical system formula_85 is defined as
where the supremum is taken over all finite measurable partitions. A theorem of Yakov Sinai in 1959 shows that the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the Bernoulli process is log 2, since almost every real number has a unique binary expansion. That is, one may partition the unit interval into the intervals [0, 1/2) and [1/2, 1]. Every real number "x" is either less than 1/2 or not; and likewise so is the fractional part of 2"n""x".
If the space "X" is compact and endowed with a topology, or is a metric space, then the topological entropy may also be defined.
Classification and anti-classification theorems.
One of the primary activities in the study of measure-preserving systems is their classification according to their properties. That is, let formula_30 be a measure space, and let formula_88 be the set of all measure preserving systems formula_1. An isomorphism formula_90 of two transformations formula_91 defines an equivalence relation formula_92 The goal is then to describe the relation formula_93. A number of classification theorems have been obtained; but quite interestingly, a number of anti-classification theorems have been found as well. The anti-classification theorems state that there are more than a countable number of isomorphism classes, and that a countable amount of information is not sufficient to classify isomorphisms.
The first anti-classification theorem, due to Hjorth, states that if formula_88 is endowed with the weak topology, then the set formula_93 is not a Borel set. There are a variety of other anti-classification results. For example, replacing isomorphism with Kakutani equivalence, it can be shown that there are uncountably many non-Kakutani equivalent ergodic measure-preserving transformations of each entropy type.
These stand in contrast to the classification theorems. These include: