Ergodic Theory. Thought of the Day 51.0

Classical dynamical systems have a particularly rich set of time symmetries. Let (X, φ) be a dynamical system. A classical dynamical system consists of a set X (the state space) and a function φ from X into itself that determines how the state changes over time (the dynamics). Let T={0,1,2,3,….}. Given any state x in X (the initial conditions), the orbit of x is the history h defined by h(0) = x, h(1) = φ(x), h(2) = φ(φ(x)), and so on. Let Ω be the set of all orbits determined by (X, φ) in this way. Let {Pr’_E}_E⊆X be any conditional probability structure on X. For any events E and D in Ω, we define Pr_E(D) = Pr’_E’(D’), where E’ is the set of all states x in X whose orbits lie in E, and D’ is the set of all states x in X whose orbits lie in D. Then {Pr_E}_E⊆Ω is a conditional probability structure on Ω. Thus, Ω and {Pr_E}_E⊆Ω together form a temporally evolving system. However, not every temporally evolving system arises in this way. Suppose the function φ (which maps from X into itself) is surjective, i.e., for all x in X, there is some y in X such that φ(y)=x. Then the set Ω of orbits is invariant under all time-shifts. Let {Pr’_E}_E⊆X be a conditional probability structure on X, and let {Pr_E}_E⊆Ω be the conditional probability structure it induces on Ω. Suppose that {Pr’_E}_E⊆X is φ-invariant, i.e., for any subsets E and D of X, if E’ = φ^–1(E) and D’ = φ^–1(D), then Pr’_E’(D’) = Pr’_E(D). Then every time shift is a temporal symmetry of the resulting temporally evolving system. The study of dynamical systems equipped with invariant probability measures is the purview of ergodic theory.

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