Ergodic Theory - Examples

Examples

• An irrational rotation of the circle R/Z, T: x → x + θ, where θ is irrational, is ergodic. This transformation has even stronger properties of unique ergodicity, minimality, and equidistribution. By contrast, if θ = p/q is rational (in lowest terms) then T is periodic, with period q, and thus cannot be ergodic: for any interval I of length a, 0 < a < 1/q, its orbit under T (that is, the union of I, T(I), …, Tq-1(I), which contains the image of I under any number of applications of T) is a T-invariant mod 0 set that is a union of q intervals of length a, hence it has measure qa strictly between 0 and 1.

• Let G be a compact abelian group, μ the normalized Haar measure, and T a group automorphism of G. Let G* be the Pontryagin dual group, consisting of the continuous characters of G, and T* be the corresponding adjoint automorphism of G*. The automorphism T is ergodic if and only if the equality (T*)n(χ)=χ is possible only when n = 0 or χ is the trivial character of G. In particular, if G is the n-dimensional torus and the automorphism T is represented by an integral matrix A then T is ergodic if and only if no eigenvalue of A is a root of unity.

• A Bernoulli shift is ergodic. More generally, ergodicity of the shift transformation associated with a sequence of i.i.d. random variables and some more general stationary processes follows from Kolmogorov's zero-one law.

• Ergodicity of a continuous dynamical system means that its trajectories "spread around" the phase space. A system with a compact phase space which has a non-constant first integral cannot be ergodic. This applies, in particular, to Hamiltonian systems with a first integral I functionally independent from the Hamilton function H and a compact level set X = {(p,q): H(p,q) = E} of constant energy. Liouville's theorem implies the existence of a finite invariant measure on X, but the dynamics of the system is constrained to the level sets of I on X, hence the system possesses invariant sets of positive but less than full measure. A property of continuous dynamical systems that is the opposite of ergodicity is complete integrability.