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.

In more detail, it is a system with the following structure: One may ask why the measure preserving transformation is defined in terms of the inverse

Now, distribute an even layer of paint on the unit interval

which preserve intersections, unions and complements (so that it is a map of Borel sets) and also sends

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

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, the answer is by stirring, mixing, turbulence, thermalization or other such processes.

The transient modes are precisely those eigenvectors of the transfer operator that have eigenvalue less than one; the invariant measure

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

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

is difficult, and, even if written down, it is hard to perform practical computations with it.

This holds in general: systems with different entropy are not isomorphic.

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.

[1] In particular, the transformations obey the rules: The earlier, simpler case fits into this framework by defining Ts = Ts for s ∈ N. The concept of a homomorphism and an isomorphism may be defined.

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.

, and let Q = {Q1, ..., Qk} be a partition of X into k measurable pair-wise disjoint sets.

Similarly, the iterated point Tnx can belong to only one of the parts as well.

, 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.

with respect to a partition Q = {Q1, ..., Qk} is then defined as Finally, the Kolmogorov–Sinai metric or measure-theoretic entropy of a dynamical system

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.

Every real number x is either less than 1/2 or not; and likewise so is the fractional part of 2nx.

One of the primary activities in the study of measure-preserving systems is their classification according to their properties.

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.

[6][7] The first anti-classification theorem, due to Hjorth, states that if

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.

, then such a generator exists iff the system is isomorphic to the Bernoulli shift on