Measure-preserving dynamical system

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 case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium.

"Area-preserving map" redirects here. For the map projection concept, see Equal-area map.

$X$ is a set,

${\mathcal {B}}$ is a over $X$ ,

σ-algebra

$\mu :{\mathcal {B}}\rightarrow [0,1]$ is a , so that $\mu (X)=1$ , and $\mu (\varnothing )=0$ ,

probability measure

$T:X\rightarrow X$ is a transformation which preserves the measure $\mu$ , i.e., $\forall A\in {\mathcal {B}}\;\;\mu (T^{-1}(A))=\mu (A)$ .

measurable

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:

Informal example[edit]

The microcanonical ensemble 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 $w\times l\times h\times \mathbb {R} ^{3}.$ A given collection of $N$ atoms would then be a single point somewhere in the space $(w\times l\times h)^{N}\times \mathbb {R} ^{3N}.$ 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, the measure $\mu$ is given by the Maxwell–Boltzmann distribution. It is a product measure, in that if $p_{i}(x,y,z,v_{x},v_{y},v_{z})\,d^{3}x\,d^{3}p$ 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 ${\mathcal {O}}\left(2^{-3N}\right).$ 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, 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.

μ could be the normalized angle measure dθ/2π on the , and T a rotation. See equidistribution theorem;

unit circle

the ;

Bernoulli scheme

the ;

interval exchange transformation

with the definition of an appropriate measure, a ;

subshift of finite type

the of a random dynamical system;

base flow

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 ) by Liouville's theorem (Hamiltonian);^[1]

symplectic volume form

for certain maps and , the Krylov–Bogolyubov theorem establishes the existence of a suitable measure to form a measure-preserving dynamical system.

Markov processes

Unlike the informal example above, the examples below are sufficiently well-defined and tractable that explicit, formal computations can be performed.

$T_{0}=\mathrm {id} _{X}:X\rightarrow X$ , the on X;

identity function

$T_{s}\circ T_{t}=T_{t+s}$ , whenever all the terms are ;

well-defined

$T_{s}^{-1}=T_{-s}$ , whenever all the terms are well-defined.

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 (or even a group, in which case we have the action of a group upon the given probability space) of transformations T_s : X → X parametrized by s ∈ Z (or R, or N ∪ {0}, or [0, +∞)), where each transformation T_s satisfies the same requirements as T above.^[1] In particular, the transformations obey the rules:

The earlier, simpler case fits into this framework by defining T_s = T^s for s ∈ N.

Generic points[edit]

A point x ∈ X is called a generic point if the orbit of the point is distributed uniformly according to the measure.

Ergodic measure-preserving transformations with a pure point spectrum have been classified.

[10]

are classified by their metric entropy.^[11]^[12]^[13] See Ornstein theory for more.

Bernoulli shifts

One of the primary activities in the study of measure-preserving systems is their classification according to their properties. That is, let $(X,{\mathcal {B}},\mu )$ be a measure space, and let $U$ be the set of all measure preserving systems $(X,{\mathcal {B}},\mu ,T)$ . An isomorphism $S\sim T$ of two transformations $S,T$ defines an equivalence relation ${\mathcal {R}}\subset U\times U.$ The goal is then to describe the relation ${\mathcal {R}}$ . 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.^[6]^[7]

The first anti-classification theorem, due to Hjorth, states that if $U$ is endowed with the weak topology, then the set ${\mathcal {R}}$ is not a Borel set.^[8] 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.^[9]

These stand in contrast to the classification theorems. These include:

on the existence of invariant measures

Krylov–Bogolyubov theorem

– Certain dynamical systems will eventually return to (or approximate) their initial state

Poincaré recurrence theorem

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). 0-19-853390-X (Provides expository introduction, with exercises, and extensive references.)

ISBN

"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 0-691-11338-6

Lai-Sang Young

T. Schürmann and I. Hoffmann, The entropy of strange billiards inside n-simplexes. J. Phys. A 28(17), page 5033, 1995. (gives a more involved example of measure-preserving dynamical system.)