IRREGULAR SETS, THE β-TRANSFORMATION AND THE ALMOST SPECIFICATION PROPERTY

Let (X, d) be a compact metric space, f : X -> X be a continuous map satisfying a property we call almost specification (which is slightly weaker than the g-almost product property of Pfister and Sullivan), and phi : X -> R be a continuous function. We show that the set of points for which the Birkhoff average of phi does not exist (which we call the irregular set) is either empty or has full topological entropy. Every beta-shift satisfies almost specification and we show that the irregular set for any beta-shift or beta-transformation is either empty or has full topological entropy and Hausdorff dimension.