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.

# Ergodic Theory. Thought of the Day 51.0

Advertisements

[…] here, here and […]