Editing Dynamical system (section)

==Ergodic systems==
{{Main|Ergodic theory}}
In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant.  This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position)&nbsp;×&nbsp;(momentum).  The flow takes points of a subset ''A'' into the points Φ<sup>&nbsp;''t''</sup>(''A'') and invariance of the phase space means that
: <math> \mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ). </math>
In the [[Hamiltonian mechanics|Hamiltonian formalism]], given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow.  The volume is said to be computed by the [[Liouville's theorem (Hamiltonian)|Liouville measure]].

In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible.  The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the [[Poincaré recurrence theorem|recurrence theorem]]: Assume the phase space has a finite Liouville volume and let ''F'' be a phase space volume-preserving map and ''A'' a subset of the phase space.  Then almost every point of ''A'' returns to ''A'' infinitely often.  The Poincaré recurrence theorem was used by [[Ernst Zermelo|Zermelo]] to object to [[Ludwig Boltzmann|Boltzmann]]'s derivation of the increase in entropy in a dynamical system of colliding atoms.

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the [[ergodic hypothesis]].  The hypothesis states that the length of time a typical trajectory spends in a region ''A'' is vol(''A'')/vol(Ω).

The ergodic hypothesis turned out not to be the essential property needed for the development of [[statistical mechanics]] and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems.  [[Bernard Koopman|Koopman]] approached the study of ergodic systems by the use of [[functional analysis]].  An observable ''a'' is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height).  The value of an observable can be computed at another time by using the evolution function φ<sup>&nbsp;t</sup>.  This introduces an operator ''U''<sup>&nbsp;''t''</sup>, the [[transfer operator]],

: <math> (U^t a)(x) = a(\Phi^{-t}(x)). </math>

By studying the spectral properties of the linear operator ''U'' it becomes possible to classify the ergodic properties of&nbsp;Φ<sup>&nbsp;''t''</sup>.  In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ<sup>&nbsp;''t''</sup> gets mapped into an infinite-dimensional linear problem involving&nbsp;''U''.

The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in [[Statistical mechanics|equilibrium statistical mechanics]].  An average in time along a trajectory is equivalent to an average in space computed with the  [[Statistical mechanics#Canonical ensemble|Boltzmann factor exp(−β''H'')]].  This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. [[SRB measure]]s replace the Boltzmann factor and they are defined on attractors of chaotic systems.