Rotations, Bernoulli shifts, and examples

Work on the circle represented by real numbers modulo one, with Lebesgue probability, and let rotation add alpha modulo one. This is an invertible bimeasurable preserving transformation for every real alpha. The Fourier characters are eigenfunctions of its Koopman operator. If alpha is irrational, no nonzero integer frequency has eigenvalue one. Expanding an invariant square-integrable function in the complete Fourier basis then leaves only its constant coefficient. Applying this to invariant indicators proves ergodicity. Completeness of the characters is part of this argument, rather than an inference from checking a few frequencies.

If alpha equals a rational fraction in lowest terms with denominator q, the qth Fourier character is nonconstant and invariant, so Lebesgue rotation is not ergodic. Its individual orbits nevertheless consist of q equally spaced points, and the uniform probability supported on any such orbit is ergodic. Thus one transformation can have different ergodic properties under different invariant measures. Irrationality eliminates finite orbits, but by itself it says nothing about decay rates for empirical averages or about independence of separated observations.

For a nonzero Fourier frequency, its spatial mean is zero and its norm is one. Its correlation with its own time shift is a unit complex number at every time. The absolute value therefore never decreases, so the bounded-observable version of weak mixing fails. In particular, an irrational circle rotation is ergodic but not weakly mixing. This is compatible with its eigenmode averages tending to zero: geometric-sum cancellation removes the signed average while leaving the absolute correlations unchanged.

Now consider bi-infinite sequences from a finite alphabet, with independent identically distributed coordinates and the product probability. The left shift reads coordinate j of the new sequence from coordinate j plus one of the old sequence. It is invertible, and identical coordinate laws make it probability preserving. A one-sided sequence shift also preserves its product probability but is generally not invertible. Mixing can be proved for either version; only claims about inverse iterates require choosing the two-sided space.

A cylinder event specifies only finitely many coordinates. Two cylinders depend on disjoint coordinate sets once one is shifted sufficiently far, so their intersection probability then equals the product exactly. This proves eventual factorization for every fixed cylinder pair. For arbitrary measurable events, approximate each in probability by cylinder events. Preservation keeps the approximation error unchanged under time shifts, and the error in intersection probabilities is bounded by the sum of the two approximation errors. Let those errors tend to zero after separating the cylinders.

This argument proves strong mixing for every measurable event pair, not just a chosen coordinate event. At overlapping times, however, cylinder events need not be independent and can even contradict one another. Checking their shared coordinate constraints before multiplying probabilities is essential. The finite modular instrument below illustrates rational orbit structure only: it has no random sequence coordinates, and a long finite cycle does not become a proof of irrational ergodicity or of Bernoulli mixing.