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Dynamical system
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===Measure theoretical definition=== {{main|Measure-preserving dynamical system}} A dynamical system may be defined formally as a measure-preserving transformation of a [[measure space]], the triplet (''T'', (''X'', Σ, ''μ''), Φ). Here, ''T'' is a monoid (usually the non-negative integers), ''X'' is a [[set (mathematics)|set]], and (''X'', Σ, ''μ'') is a [[measure space|probability space]], meaning that Σ is a [[sigma-algebra]] on ''X'' and μ is a finite [[measure (mathematics)|measure]] on (''X'', Σ). A map Φ: ''X'' → ''X'' is said to be [[measurable function|Σ-measurable]] if and only if, for every σ in Σ, one has <math>\Phi^{-1}\sigma \in \Sigma</math>. A map Φ is said to '''preserve the measure''' if and only if, for every ''σ'' in Σ, one has <math>\mu(\Phi^{-1}\sigma ) = \mu(\sigma)</math>. Combining the above, a map Φ is said to be a '''measure-preserving transformation of ''X'' ''', if it is a map from ''X'' to itself, it is Σ-measurable, and is measure-preserving. The triplet (''T'', (''X'', Σ, ''μ''), Φ), for such a Φ, is then defined to be a '''dynamical system'''. The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the [[iterated function|iterates]] <math>\Phi^n = \Phi \circ \Phi \circ \dots \circ \Phi</math> for every integer ''n'' are studied. For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated. ====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.
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