A class of compact group extensions of β-transformations

Abstract

Skew product Compact group extension Central Limit TheoremWe investigate the dynamical properties of a skew product transformation T ϕ on [0, 1) × G defined by T ϕ (x, g) = (T x, g · ϕ(x)) where T is the β-transformation for β 2 and ϕ(x) is a compact group G-valued step function with a finite number of discontinuities. We give several sufficient conditions for ergodicity and strong mixing of T ϕ . As an application, we describe a class of step functions which satisfy the Central Limit Theorem for the β-transformations. As another application, we also consider a class of skew product transformations T β,a,w on [0, 1) × [0, 1) which maps (x, y) → (βx, y + ax + w) (mod 1) where a, w ∈ R and give necessary and sufficient conditions for ergodicity and strong mixing.

Normal Numbers Mod 2 on the Logistic Map

“…In [1,2], the authors consider the case when transformations defined by x → Lx (mod 1) with L ∈ R on X = [0, 1) and show that the sequence {d n } is evenly distributed if exp(πi1 E (x)) has finite L-adic discontinuity points…”

“…Recall that the transformations which are considered in [1,2,6] are Lebesgue measure preserving or its invariant density function is bounded. Contrast to those, the logistic map does not preserve Lebesgue measure and also its invariant density function is not bounded.…”