 Measurepreserving dynamical system

In mathematics, a measurepreserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.
Contents
Definition
A 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:
 X is a set,
 is a σalgebra over X,
 is a probability measure, so that μ(X) = 1, and
 is a measurable transformation which preserves the measure μ, i. e. each satisfies
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 identity function on X;
 , whenever all the terms are welldefined;
 , whenever all the terms are welldefined.
The earlier, simpler case fits into this framework by defining for .
The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem.
Examples
Examples include:
 μ could be the normalized angle measure dθ/2π on the unit circle, and T a rotation. See equidistribution theorem;
 the Bernoulli scheme;
 with the definition of an appropriate measure, a subshift of finite type;
 the base flow of a random dynamical system.
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 map φ is measurable,
 For each , one has μ(ϕ ^{− 1}B) = ν(B),
 For μalmost all , one has ϕ(Tx) = S(ϕx).
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
 For μalmost all , one has x = ψ(ϕx)
 For νalmost all , one has y = ϕ(ψy).
Generic points
A point 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 , and let Q = { Q_{1}, ..., Q_{k} } be a partition of X into k measurable pairwise disjoint pieces. Given a point x ∈ X, 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 partitions
Given a partition Q = { Q_{1}, ..., Q_{k} } and a dynamical system , we define Tpullback of Q as
Further, given two partitions Q = { Q_{1}, ..., Q_{k} } and R = { R_{1}, ..., 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 measuretheoretic entropy of a dynamical system.
Measuretheoretic entropy
The entropy of a partition Q is defined as
The measuretheoretic entropy of a dynamical system with respect to a partition Q = { Q_{1}, ..., Q_{k} } is then defined as
Finally, the Kolmogorov–Sinai or measuretheoretic entropy of a dynamical system is defined as
where the supremum is taken over all finite measurable partitions. A theorem of Yakov G. 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 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 compact and endowed with a topology, or is a metric space, then the topological entropy may also be defined.
See also
 Krylov–Bogolyubov theorem on the existence of invariant measures
 Poincaré recurrence theorem
References
 Michael S. Keane, Ergodic theory and subshifts of finite type, (1991), appearing as Chapter 2 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). ISBN 019853390X (Provides expository introduction, with exercises, and extensive references.)
 LaiSang Young, "Entropy in Dynamical Systems" (pdf; ps), appearing as Chapter 16 in Entropy, Andreas Greven, Gerhard Keller, and Gerald Warnecke, eds. Princeton University Press, Princeton, NJ (2003). ISBN 0691113386
Examples
 T. Schürmann and I. Hoffmann, The entropy of strange billiards inside nsimplexes. J. Phys. A28, page 5033ff, 1995. PDFDokument
Categories: Dynamical systems
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