Rational ergodicity of step function skew products

Abstract

We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk. §0 Introduction A rational step function is a right continuous, step function on the additive circle T ∶= R Z ≅ [0, 1) taking values in R d , whose discontinuity points are rational.Let ϕ ∶ T → R d be a rational step function.The skew productsT α,ϕ (x, y) ∶= (x + α, y + ϕ(x)) are conservative if and only if T ϕ(t)dt = 0.Necessity follows from the ergodic theorem and sufficiency follows from the Denjoy-Koksma inequality (see below).Consider the collections of badly approximable irrationals

A temporal central limit theorem for real-valued cocycles over rotations

“…In the same context (substitutions with eigenvalues of modulus one), Paquette and Son [28] recently also proved a temporal CLT. In [1] a temporal CLT over quadratic irrational rotations and R d valued, piecewise constant functions with rational discontinuities, is shown to hold along subsequences.…”