1.8 KiB
Topic 5: Introducing dynamics: classical and non-commutative
Section 1: Dynamics in classical probability
Basic definitions
Definition of orbit
Let T:\Omega\to\Omega be a map (may not be invertible) generating a dynamical system on \Omega. Given \omega\in \Omega, the (forward) orbit of \omega is the set \mathscr{O}(\omega)=\{T^n(\omega)\}_{n\in\mathbb{Z}}.
The theory of dynamics is the study of properties of orbits.
Definition of measure-preserving map
Let P be a probability measure on a $\sigma$-algebra \mathscr{F} of subsets of \Omega. A measurable transformation T:\Omega\to\Omega is said to be measure-preserving if for all random variables \psi:\Omega\to\mathbb{R}, we have \mathbb{E}(\psi\circ T)=\mathbb{E}(\psi), that is:
\int_\Omega (\psi\circ T)(\omega)dP(\omega)=\int_\Omega \psi(\omega)dP(\omega)
Definition of ergodic map
A measurable transformation T:\Omega\to\Omega is said to be ergodic if for all random variables \psi:\Omega\to\mathbb{R}, we have \mathbb{E}(\psi\circ T)=\mathbb{E}(\psi), that is:
\int_\Omega (\psi\circ T)(\omega)dP(\omega)=\int_\Omega \psi(\omega)dP(\omega)
Definition of isometry
The composition operator \psi\mapsto U\psi=\psi\circ T, where T is a measure preserving map defined on \mathscr{H}=L^2(\Omega,\mathscr{F},P) is isometry of \mathscr{H} if \langle U\psi,U\phi\rangle=\langle\psi,\phi\rangle for all \psi,\phi\in\mathscr{H}.
Definition of unitary
The composition operator \psi\mapsto U\psi=\psi\circ T, where T is a measure preserving map defined on \mathscr{H}=L^2(\Omega,\mathscr{F},P) is unitary of \mathscr{H} if U is an isometry and T is invertible with measurable inverse.