Measure preserving dynamical system

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:

This definition 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) of transformations T_s : X → X parametrized by s ∈ Z (or R, or N ∪ {0}, or [0, +∞)), where each transformation T_s satisfies the same requirements as T above. In particular, the transformations obey the rules:

The earlier, simpler case fits into this framework by definingT_s = T^s for s ∈ N.

The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem.

Examples

Examples include:

Homomorphisms

The concept of a homomorphism and an isomorphism may be defined.

Consider two dynamical systems ( X , A , μ , T ) and ( Y , B , ν , S ) . Then a mapping

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

1. The map φ is measurable,
2. For each B ∈ B , one has μ ( φ − 1 B ) = ν ( B ) ,
3. For μ-almost all x ∈ X, one has φ(Tx) = S(φ x).

The system ( Y , B , ν , S ) is then called a factor of ( X , A , μ , T ) .

The map φ is an isomorphism of dynamical systems if, in addition, there exists another mapping

that is also a homomorphism, which satisfies

1. For μ-almost all x ∈ X, one has x = ψ ( φ x )
2. For ν-almost all y ∈ Y, one has y = φ ( ψ y ) .

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 ( X , B , T , μ ) , and let Q = {Q₁, ..., Q_k} 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 Q_i. 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 = {Q₁, ..., Q_k} and a dynamical system ( X , B , T , μ ) , we define T-pullback of Q as

Further, given two partitions Q = {Q₁, ..., Q_k} and R = {R₁, ..., R_m}, we define their refinement as

With these two constructs we may define refinement of an iterated pullback

which plays crucial role in the construction of the measure-theoretic entropy of a dynamical system.

Measure-theoretic entropy

The entropy of a partition Q is defined as

The measure-theoretic entropy of a dynamical system ( X , B , T , μ ) with respect to a partition Q = {Q₁, ..., Q_k} is then defined as

Finally, the Kolmogorov–Sinai or metric or measure-theoretic entropy of a dynamical system ( X , B , T , μ ) 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 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.

Examples