Normal Points for Generic Hyperbolic Maps

Given a continuous transformation on a compact space we call a point normal if the ergodic averages of any continuous functions converge. For example, the familiar notion of a normal number 0 < ξ < 1 in the context of number theory is one which is normal in the above sense for all of the transformations T : [0, 1) → [0, 1) given by T (ξ) = dξ (mod 1) for each d ≥ 2. In this note, we want to consider the question of when a specific point is normal for typical transformations, in some suitable sense. We begin by considering a particularly simple setting. Let f(λ) : K → K, for λ ∈ (−#, #), be a family of C1 orientation preserving expanding maps of the unit circle K = R/Z of degree d ≥ 2 which are perturbations of the standard linear map f(0) : K → K given by f(0)(ξ) = dξ (mod 1). A simple result is the following.