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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. (that is, P:\mathscr{F}\to anything) 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)
Example:
The doubling map T:\Omega\to\Omega is defined as T(x)=2x\mod 1, is a Lebesgue measure preserving map on \Omega=[0,1].
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.
Section 2: Continuous time (classical) dynamical systems
Spring-mass system
The pure state of the system is given by the position and velocity of the mass. (x,v) is a point in \mathbb{R}^2. \mathbb{R}^2 is the state space of the system. (or phase space)
The motion of the system in its state space is a closed curve.
\Phi_t(x,v)=\left(\cos(\omega t)x-\frac{1}{\omega}\sin(\omega t)v, \cos(\omega t)v-\omega\sin(\omega t)x\right)
Such system with closed curve is called integrable system. Where the doubling map produces orbits having distinct dynamical properties (chaotic system).
Note, some section is intentionally ignored here. They are about in the setting of operators on Hilbert spaces, the evolution of (classical, non-dissipative e.g. linear spring-mass system) system, is implemented by a one-parameter group of unitary operators.
The detailed construction is omitted here.
Definition of Hermitian operator
A linear operator A on a Hilbert space \mathscr{H} is said to be Hermitian if \forall \psi,\phi\in domain of $A$, we have \langle A\psi,\phi\rangle=\langle\psi,A\phi\rangle.
It is skew-Hermitian if \langle A\psi,\phi\rangle=-\langle\psi,A\phi\rangle.