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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a measure-preserving dynamical system is an object of study in the abstract formulation of
dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
, and
ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of
conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink ...
s. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
(in particular, most non-dissipative systems) as well as systems in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
.


Definition

A measure-preserving dynamical system is defined as a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
and a
measure-preserving In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...
transformation on it. In more detail, it is a system :(X, \mathcal, \mu, T) with the following structure: *X is a set, *\mathcal B is a σ-algebra over X, *\mu:\mathcal\rightarrow ,1/math> is a
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''. The difference between a probability measure and the more gener ...
, so that \mu (X) = 1, and \mu(\varnothing) = 0, * T:X \rightarrow X is a
measurable In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
transformation which
preserves Fruit preserves are preparations of fruits whose main preserving agent is sugar and sometimes acid, often stored in glass jars and used as a condiment or spread. There are many varieties of fruit preserves globally, distinguished by the method ...
the measure \mu, i.e., \forall A\in \mathcal\;\; \mu(T^(A))=\mu(A) .


Discussion

One may ask why the measure preserving transformation is defined in terms of the inverse \mu(T^(A))=\mu(A) instead of the forward transformation \mu(T(A))=\mu(A). This can be understood in a fairly easy fashion. Consider a mapping \mathcal of
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
s: :\mathcal:P(X)\to P(X) Consider now the special case of maps \mathcal which preserve intersections, unions and complements (so that it is a map of
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
s) and also sends X to X (because we want it to be
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
). Every such conservative, Borel-preserving map can be specified by some
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
map T:X\to X by writing \mathcal(A)=T^(A). Of course, one could also define \mathcal(A)=T(A), but this is not enough to specify all such possible maps \mathcal. That is, conservative, Borel-preserving maps \mathcal cannot, in general, be written in the form \mathcal(A)=T(A); one might consider, for example, the map of the unit interval T: ,1)_\to_[0,1)_given_by_x_\mapsto_2x\mod_1;_this_is_the_Bernoulli_map. Note_that_\mu(T^(A))_has_the_form_of_a_Pushforward_measure.html" "title="Bernoulli_map.html" ;"title=",1) \to [0,1) given by x \mapsto 2x\mod 1; this is the Bernoulli map">,1) \to [0,1) given by x \mapsto 2x\mod 1; this is the Bernoulli map. Note that \mu(T^(A)) has the form of a Pushforward measure">pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
, whereas \mu(T(A)) is generically called a pullback. Almost all properties and behaviors of dynamical systems are defined in terms of the pushforward. For example, the
transfer operator Transfer may refer to: Arts and media * ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović * ''Transfer'' (1966 film), a short film * ''Transfer'' (journal), in management studies ...
is defined in terms of the pushforward of the transformation map T; the measure \mu can now be understood as an
invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, ...
; it is just the Frobenius–Perron eigenvector of the transfer operator (recall, the FP eigenvector is the largest eigenvector of a matrix; in this case it is the eigenvector which has the eigenvalue one: the invariant measure.) There are two classification problems of interest. One, discussed below, fixes (X, \mathcal, \mu) and asks about the isomorphism classes of a transformation map T. The other, discussed in
transfer operator Transfer may refer to: Arts and media * ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović * ''Transfer'' (1966 film), a short film * ''Transfer'' (journal), in management studies ...
, fixes (X, \mathcal) and T, and asks about maps \mu that are measure-like. Measure-like, in that they preserve the Borel properties, but are no longer invariant; they are in general dissipative and so give insights into
dissipative system A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dis ...
s and the route to equilibrium. In terms of physics, the measure-preserving dynamical system (X, \mathcal, \mu, T) often describes a physical system that is in equilibrium, for example,
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
. One might ask: how did it get that way? Often, the answer is by stirring, mixing,
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
,
thermalization In physics, thermalisation is the process of physical bodies reaching thermal equilibrium through mutual interaction. In general the natural tendency of a system is towards a state of equipartition of energy and uniform temperature that maximizes ...
or other such processes. If a transformation map T describes this stirring, mixing, etc. then the system (X, \mathcal, \mu, T) is all that is left, after all of the transient modes have decayed away. The transient modes are precisely those eigenvectors of the transfer operator that have eigenvalue less than one; the invariant measure \mu is the one mode that does not decay away. The rate of decay of the transient modes are given by (the logarithm of) their eigenvalues; the eigenvalue one corresponds to infinite half-life.


Informal example

The
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
from physics provides an informal example. Consider, for example, a fluid, gas or plasma in a box of width, length and height w\times l\times h, consisting of N atoms. A single atom in that box might be anywhere, having arbitrary velocity; it would be represented by a single point in w\times l\times h\times \mathbb^3. A given collection of N atoms would then be a ''single point'' somewhere in the space (w\times l\times h)^N \times \mathbb^. The "ensemble" is the collection of all such points, that is, the collection of all such possible boxes (of which there are an uncountably-infinite number). This ensemble of all-possible-boxes is the space X above. In the case of an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
, the measure \mu is given by the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used ...
. It is a
product measure In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two ...
, in that if p_i(x,y,z,v_x,v_y,v_z)\,d^3x\,d^3p is the probability of atom i having position and velocity x,y,z,v_x,v_y,v_z, then, for N atoms, the probability is the product of N of these. This measure is understood to apply to the ensemble. So, for example, one of the possible boxes in the ensemble has all of the atoms on one side of the box. One can compute the likelihood of this, in the Maxwell–Boltzmann measure. It will be enormously tiny, of order \mathcal\left(2^\right). Of all possible boxes in the ensemble, this is a ridiculously small fraction. The only reason that this is an "informal example" is because writing down the transition function T is difficult, and, even if written down, it is hard to perform practical computations with it. Difficulties are compounded if the interaction is not an ideal-gas billiard-ball type interaction, but is instead a
van der Waals interaction In molecular physics, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and the ...
, or some other interaction suitable for a liquid or a plasma; in such cases, the invariant measure is no longer the Maxwell–Boltzmann distribution. The art of physics is finding reasonable approximations. This system does exhibit one key idea from the classification of measure-preserving dynamical systems: two ensembles, having different temperatures, are inequivalent. The entropy for a given canonical ensemble depends on its temperature; as physical systems, it is "obvious" that when the temperatures differ, so do the systems. This holds in general: systems with different entropy are not isomorphic.


Examples

Unlike the informal example above, the examples below are sufficiently well-defined and tractable that explicit, formal computations can be performed. * μ could be the normalized angle measure dθ/2π on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
, and ''T'' a rotation. See
equidistribution theorem In mathematics, the equidistribution theorem is the statement that the sequence :''a'', 2''a'', 3''a'', ... mod 1 is uniformly distributed on the circle \mathbb/\mathbb, when ''a'' is an irrational number. It is a special case of the ergodic ...
; * the
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 appear naturally in symbolic dynamics, and are thus important in the study of dynamical sy ...
; * the
interval exchange transformation In mathematics, an interval exchange transformation is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and ...
; * with the definition of an appropriate measure, a
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 In the mathematical field of dynamical systems, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are characterized by a state space ''S'', a set of ma ...
; * the flow of a Hamiltonian vector field on the tangent bundle of a closed connected smooth manifold is measure-preserving (using the measure induced on the Borel sets by the symplectic volume form) by
Liouville's theorem (Hamiltonian) In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajectorie ...
; * for certain maps and
Markov processes Markov ( Bulgarian, russian: Марков), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include: Academics *Ivana Markova (born 1938), Czechoslovak-British emeritus professor of psychology at ...
, the
Krylov–Bogolyubov theorem In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical systems. The theorems guarantee the existence of i ...
establishes the existence of a suitable measure to form a measure-preserving dynamical system.


Generalization to groups and monoids

The definition of a measure-preserving dynamical system 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 In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
(or even a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, in which case we have the action of a group upon the given probability space) of transformations ''Ts'' : ''X'' → ''X'' parametrized by ''s'' ∈ Z (or R, or N ∪ , or [0, +∞)), where each transformation ''Ts'' satisfies the same requirements as ''T'' above. In particular, the transformations obey the rules: * T_0 = \mathrm_X :X \rightarrow X, the identity function on ''X''; * T_ \circ T_ = T_, whenever all the terms are well-defined; * T_^ = T_, whenever all the terms are well-defined. The earlier, simpler case fits into this framework by defining ''Ts'' = ''Ts'' for ''s'' ∈ N.


Homomorphisms

The concept of a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
and an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
may be defined. Consider two dynamical systems (X, \mathcal, \mu, T) and (Y, \mathcal, \nu, S). Then a mapping :\varphi:X \to Y is a homomorphism of dynamical systems if it satisfies the following three properties: # The map \varphi\ is
measurable In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
. # For each B \in \mathcal, one has \mu (\varphi^B) = \nu(B). # For \mu-almost all x \in X, one has \varphi(Tx) = S(\varphi x). The system (Y, \mathcal, \nu, S) is then called a factor of (X, \mathcal, \mu, T). The map \varphi\; is an isomorphism of dynamical systems if, in addition, there exists another mapping :\psi:Y \to X that is also a homomorphism, which satisfies # for \mu-almost all x \in X, one has x = \psi(\varphi x); # for \nu-almost all y \in Y, one has y = \varphi(\psi y). Hence, one may form a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
of dynamical systems and their homomorphisms.


Generic points

A point ''x'' ∈ ''X'' is called a generic point if the
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
of the point is distributed uniformly according to the measure.


Symbolic names and generators

Consider a dynamical system (X, \mathcal, T, \mu), and let ''Q'' = be a
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
of ''X'' into ''k'' measurable pair-wise disjoint pieces. Given a point ''x'' ∈ ''X'', clearly ''x'' belongs 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 :T^nx \in Q_. The set of symbolic names with respect to a partition is called the
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 (e ...
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 = and a dynamical system (X, \mathcal, T, \mu), define the ''T''-pullback of ''Q'' as : T^Q = \. Further, given two
partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
''Q'' = and ''R'' = , define their
refinement Refinement may refer to: Mathematics * Equilibrium refinement, the identification of actualized equilibria in game theory * Refinement of an equivalence relation, in mathematics ** Refinement (topology), the refinement of an open cover in mathem ...
as : Q \vee R = \. With these two constructs, the ''refinement of an iterated pullback'' is defined as : \begin \bigvee_^N T^Q & = \ \\ \end which plays crucial role in the construction of the measure-theoretic entropy of a dynamical system.


Measure-theoretic entropy

The
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
of a partition \mathcal is defined as :H(\mathcal)=-\sum_\mu (Q) \log \mu(Q). The measure-theoretic entropy of a dynamical system (X, \mathcal, T, \mu) with respect to a partition ''Q'' = is then defined as :h_\mu(T,\mathcal) = \lim_ \frac H\left(\bigvee_^N T^\mathcal\right). Finally, the Kolmogorov–Sinai metric or measure-theoretic entropy of a dynamical system (X, \mathcal,T,\mu) is defined as :h_\mu(T) = \sup_Q h_\mu(T,Q). where the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
is taken over all finite measurable partitions. A theorem of
Yakov Sinai Yakov Grigorevich Sinai (russian: link=no, Я́ков Григо́рьевич Сина́й; born September 21, 1935) is a Russian-American mathematician known for his work on dynamical systems. He contributed to the modern metric theory of dy ...
in 1959 shows that the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the
Bernoulli process In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) 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. Th ...
is log 2, since
almost every In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
has a unique
binary expansion A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...
. That is, one may partition the
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. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
into the intervals /nowiki>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.


Classification and anti-classification theorems

One of the primary activities in the study of measure-preserving systems is their classification according to their properties. That is, let (X, \mathcal, \mu) be a measure space, and let U be the set of all measure preserving systems (X, \mathcal, \mu, T). An isomorphism S\sim T of two transformations S, T defines an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
\mathcal\subset U\times U. The goal is then to describe the relation \mathcal. A number of classification theorems have been obtained; but quite interestingly, a number of anti-classification theorems have been found as well. The anti-classification theorems state that there are more than a countable number of isomorphism classes, and that a countable amount of information is not sufficient to classify isomorphisms. The first anti-classification theorem, due to Hjorth, states that if U is endowed with the
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
, then the set \mathcal is not a
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
. There are a variety of other anti-classification results. For example, replacing isomorphism with Kakutani equivalence, it can be shown that there are uncountably many non-Kakutani equivalent ergodic measure-preserving transformations of each entropy type. These stand in contrast to the classification theorems. These include: * Ergodic measure-preserving transformations with a pure point spectrum have been classified. *
Bernoulli shift In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical syst ...
s are classified by their metric entropy. See
Ornstein theory In mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic. The result, given by Donald Ornstein in 1970, is important ...
for more.


See also

*
Krylov–Bogolyubov theorem In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical systems. The theorems guarantee the existence of i ...
on the existence of invariant measures * Poincaré recurrence theorem


References


Further reading

* 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). ''(Provides expository introduction, with exercises, and extensive references.)'' *
Lai-Sang Young Lai-Sang Lily Young (, born 1952) is a Hong Kong-born American mathematician who holds the Henry & Lucy Moses Professorship of Science and is a professor of mathematics and neural science at the Courant Institute of Mathematical Sciences of New ...
, "Entropy in Dynamical Systems"
pdfps
, 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 * T. Schürmann and I. Hoffmann, ''The entropy of strange billiards inside n-simplexes.'' J. Phys. A 28(17), page 5033, 1995
PDF-Document
''(gives a more involved example of measure-preserving dynamical system.)'' Dynamical systems Entropy and information Measure theory Entropy Information theory