Editing Dynamical system (section)

====Relation to geometric definition====

The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the [[Krylov–Bogolyubov theorem]]) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.

Some systems have a natural measure, such as the [[Liouville's theorem (Hamiltonian)|Liouville measure]] in [[Hamiltonian system]]s, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system.  For chaotic [[dissipative system]]s the choice of invariant measure is technically more challenging. The measure needs to be supported on the [[attractor]], but attractors have zero [[Lebesgue measure]] and the invariant measures must be singular with respect to the Lebesgue measure. A small region of phase space shrinks under time evolution.

For hyperbolic dynamical systems, the [[Sinai–Ruelle–Bowen measure]]s appear to be the natural choice.  They are constructed on the geometrical structure of [[stable manifold|stable and unstable manifold]]s of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.