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Measurepreserving dynamical systemIn mathematics, a measurepreserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Additional recommended knowledge
DefinitionA measurepreserving dynamical system is defined as a probability space and a measurepreserving 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 parametrized by (or , or , or ), 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 defining T_{s}: = T^{s} for . ExamplesExamples include:
DiscussionOne may wonder why the seemingly simpler identity
is not used. Here is the problem: suppose T : [0, 1] → [0, 1] is defined by T(x) = (4x mod 1), i.e., T(x) is the "fractional part" of 4x. Then the interval [0.01, 0.02] is mapped to an interval four times as long as itself, but nonetheless the measure of T^{ −1}( [0.04, 0.08] ) = [0.01, 0.02] ∪ [0.26, 0.27] ∪ [0.51, 0.52] ∪ [0.76, 0.77] is no different from the measure of [0.04, 0.08]. That hypothesis suffices for the proofs of ergodic theorems. This transformation is measurepreserving. HomomorphismsThe 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 φ is an isomorphism of dynamical systems if, in addition, there exists another mapping that is also a homomorphism, which satisfies
Generic pointsA point is called a generic point if the orbit of the point is distributed uniformly according to the measure. Symbolic names and generatorsLet be a partition of X into k measurable pairwise disjoint pieces. Given a point , clearly x belongs to only one of the Q_{i}. Similarly, the iterated point T^{n}x 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 μalmost every point x has a unique symbolic name. Operations on partitionsGiven a partition and a dynamical system , we define Tpullback of Q as Further, given two partitions and , 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 measuretheoretic entropy of a dynamical system. Measuretheoretic entropyThe entropy of a partition Q is defined as The measuretheoretic entropy of a dynamical system with respect to a partition is then defined as Finally, the measuretheoretic entropy of a dynamical system is defined as where the supremum is taken over all finite measurable partitions. A theorem of Ya. 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 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 endowed with a metric, then the topological entropy may also be defined. References


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Measurepreserving_dynamical_system". A list of authors is available in Wikipedia. 