Three Lectures given in Torun On: Selected Topics in Modern Ergodic Theory

Example (1.1). We will start with some examples of dynamical systems: • S with Lebesgue σ-algebra and normalised measure, and rotation as the transformation; • multidimensional tori with the product σ-algebra, product measure and transformation given by an algebraic automorphism. (Lebesgue measure is preserved by unicity of the Haar measure). • for dynamical systems, the Hamiltonian flow ( ∂p = −ṗ, ∂H ∂q = q̇) equipped with the Liouville measure dp.dq which it leaves invariant (finite when the phase space is compact). • consider a probability space and a bi-infinite sequence (Xn) of random variables such that, for all n, k > 0, the law of (Xn, Xn+1, ..Xn+k) is independent of k; (this is called a stationary stochastic process) then the shift transformation Θ(Xn)n∈Z := (Xn+1)n∈Z also generates a dynamical system. Of particular importance are the examples where the (Xn) form an independent family and where the common law of the Xn is a finite probability vector π. The corresponding dynamical system is the Bernoulli shift B(π).