In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory

Ergodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

 in particular.

Definition

A measure-preserving dynamical system is defined as a probability space

Probability space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

 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
  Sigma-algebra
  In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
  
   over ,
• is a probability measure
  Probability measure
  In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
  
  , so that , and
• is a measurable
  Measurable function
  In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
  
   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

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

 (or even a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

) 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
  Identity function
  In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
  
   on ;
• , whenever all the terms are well-defined
  Well-defined
  In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...
  
  ;
• , whenever all the terms are well-defined.

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
  Unit circle
  In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
  
  , and a rotation. See equidistribution theorem
  Equidistribution theorem
  In mathematics, the equidistribution theorem is the statement that the sequenceis uniformly distributed on the unit interval, when a is an irrational number...
  
  ;

• the Bernoulli scheme
  Bernoulli scheme
  In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes are important in the study of dynamical systems, as most such systems exhibit a repellor that is the product of the Cantor set and a smooth...
  
  ;

• the interval exchange transformation;

• with the definition of an appropriate measure, a subshift of finite type
  Subshift of finite type
  In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite state machine...
  
  ;

• the base flow of a random dynamical system
  Random dynamical system
  In mathematics, a random dynamical system is a measure-theoretic formulation of a dynamical system with an element of "randomness", such as the dynamics of solutions to a stochastic differential equation...
  
  .

Homomorphisms

The concept of a homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

 and an isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 may be defined.

Consider two dynamical systems and . Then a mapping


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

1. The map φ is measurable
   Measurable function
   In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
   
   ,
2. For each , one has ,
3. For μ-almost all
   Almost everywhere
   In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
   
    , 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

1. For μ-almost all , one has
2. For ν-almost all , one has .

Generic points

A point is called a generic point if the orbit

Orbit (dynamics)

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...

 of the point is distributed uniformly according to the measure.

Symbolic names and generators

Consider a dynamical system , and let Q = { Q₁, ..., Q_k } be a partition

Partition of a set

In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...

 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 T ⁿx 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

Symbolic dynamics

In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics given by the shift operator...

 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 , 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

Information entropy

In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits...

 of a partition Q is defined as


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


Finally, the Kolmogorov–Sinai or measure-theoretic entropy of a dynamical system is defined as


where the supremum

Supremum

In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

 is taken over all finite measurable partitions. A theorem of Yakov G. Sinai

Yakov G. Sinai

Yakov Grigorevich Sinai is an influential mathematician working in the theory of dynamical systems, in mathematical physics and in probability theory. His work has shaped the modern metric theory of dynamical systems...

 in 1959 shows that the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the Bernoulli process

Bernoulli process

In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables Xi are identical and independent...

 is log 2, since every real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 has a unique binary expansion. That is, one may partition the unit interval

Unit interval

In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...

 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

Topological entropy

In mathematics, the topological entropy of a topological dynamical system is a nonnegative real number that measures the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the...

 may also be defined.

See also

• Krylov–Bogolyubov theorem on the existence of invariant measures
• Poincaré recurrence theorem
  Poincaré recurrence theorem
  In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which...
  
  

Examples

• T. Schürmann and I. Hoffmann, The entropy of strange billiards inside n-simplexes. J. Phys. A28, page 5033ff, 1995. PDF-Dokument

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