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 of transformations Ts : X → X parametrized by s ∈ Z ), 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.
Examples
Homomorphisms
The concept of a homomorphism and an isomorphism may be defined. Consider two dynamical systems and. Then a mapping is a homomorphism of dynamical systems if it satisfies the following three properties:
The system is then called a factor of. The map is an isomorphism of dynamical systems if, in addition, there exists another mapping that is also a homomorphism, which satisfies
for μ-almost all, one has ;
for ν-almost all, one has.
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.
Consider a dynamical system, and letQ = be a partition of X into k measurable pair-wise disjoint pieces. Given a point x ∈ X, clearly xbelongs 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 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.
Given a partition Q = and a dynamical system, we define T-pullback of Q as Further, given two partitions Q = and R =, 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 with respect to a partition Q = is then defined as Finally, the Kolmogorov–Sinai metric or measure-theoretic entropy of a dynamical system 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 everyreal number has a unique binary expansion. That is, one may partition the unit interval into the intervals. Every real number x is either less than 1/2 or not; and likewise so is the fractional part of 2nx. If the spaceX is compact and endowed with a topology, or is a metric space, then the topological entropy may also be defined.
