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Wikipedia / 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:

is a set,
is a σ-algebra,
is a probability measure,
is a measurable transformation which preserves.

Discussion

. This can be understood intuitively.

. This is the Bernoulli map half as well. The two layers of thin paint, layered together, recreates the exact same paint thickness.

of power sets:

which preserve intersections, unions and complements (so that it is a map of Borel sets (because we want it to be conservative). Every such conservative, Borel-preserving map can be specified by some surjective.

has the form of a pushforward 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 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.)

. The other, discussed in transfer operator 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.

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 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 atoms would then be a single point above.

In the case of an ideal gas is given by the Maxwell–Boltzmann distribution. It is a product measure Of all possible boxes in the ensemble, this is a ridiculously small fraction.

is difficult, and, even if written down, it is hard to perform practical computations with it. Difficulties are compounded if there are interactions between the particles themselves, like 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.

μ could be the normalized angle measure dθ/2π on the unit circle, and T a rotation. See equidistribution theorem;
the Bernoulli scheme;
the interval exchange transformation;
with the definition of an appropriate measure, a subshift of finite type;
the base flow of a random dynamical system;
the flow of a Hamiltonian vector field on the tangent bundle of a closed connected smooth manifold is measure-preserving (using the measure induced on the Borel sets by the symplectic volume form) by Liouville's theorem (Hamiltonian);
for certain maps and Markov processes, the Krylov–Bogolyubov theorem establishes the existence of a suitable measure to form a measure-preserving dynamical system.

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 identity function on X;
, whenever all the terms are well-defined;
, whenever all the terms are well-defined.

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.

. Then a mapping

is a homomorphism of dynamical systems if it satisfies the following three properties:

is then called a factor.

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

, and let Q = {Q1, ..., Qk} be a partition of X into k measurable pair-wise disjoint sets. 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 {an} 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 = {Q1, ..., Qk, define the T-pullback of Q as

Further, given two partitions Q = {Q1, ..., Qk} and R = {R1, ..., Rm}, 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 is defined as

with respect to a partition Q = {Q1, ..., Qk} is then defined as

Finally, the Kolmogorov–Sinai metric or measure-theoretic entropy 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 log2, since almost every real number has a unique binary expansion. That is, one may partition the unit interval0, and [1/2,1]. Every real number x is either less than 1/2 or not; and likewise so is the fractional part of 2nx.

If the space X is compact and endowed with a topology, or is a metric space, then the topological entropy may also be defined.

is absolutely continuous with respect to the Lebesgue measure, then we have the Rokhlin formula (section 4.3 and section 12.3 This allows calculation of entropy of many interval maps, such as the logistic map.

Classification and anti-classification theorems

defines an equivalence relation. 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.

is endowed with the weak topology 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:

Ergodic measure-preserving transformations with a pure point spectrum have been classified.
Bernoulli shifts are classified by their metric entropy. See Ornstein theory for more.