In
mathematics, a
measure-preserving dynamical system is an object of study in the abstract formulation of
dynamical systems, and
ergodic theory in particular.
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:
- is a set,
- is a σ-algebra over ,
- is a probability measure, so that , and
- is a measurable transformation which preserves the measure , i. e. each satisfies
This definition can be generalized to the case in which 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 satisfies the same requirements as above. In particular, the transformations obey the rules
- , the identity function on ;
- , whenever all the terms are well-defined;
- , whenever all the terms are well-defined.
The earlier, simpler case fits into this framework by defining for .
Examples
Examples include:
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 ,
- For μ-almost all , one has .
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 μ-almost all , one has
- For ν-almost all , one has .
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
Let
be a
partition of
X into
k measurable pair-wise disjoint pieces. Given a point
, clearly
x belongs to only one of the
. Similarly, the iterated point
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.
Operations on partitions
Given a partition and a dynamical system , we define
-pullback of 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 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 is then defined as
Finally, the Kolmogorov–Sinai or measure-theoretic 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 , since every real number has a unique binary expansion. That is, one may partition the unit interval into the intervals and . Every real number x is either less than 1/2 or not; and likewise so is the fractional part of .
If the space X is endowed with a metric, then the topological entropy may also be defined.
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 0-19-853390-X (Provides expository introduction, with exercises, and extensive references.)
- Lai-Sang Young, "Entropy in Dynamical Systems", appearing as Chapter 16 in Entropy, Andreas Greven, Gerhard Keller, and Gerald Warnecke, eds. Princeton University Press, Princeton, NJ (2003). ISBN 0-691-11338-6
Examples
- T. Schürmann and I. Hoffmann, The entropy of strange billiards inside n-simplexes. J. Phys. A28, page 5033ff, 1995. PDF-Dokument