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

Let (X, d) be a compact metric space, ƒ: 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 φ : X ⟼ ℝ be a continuous function. We show that the set of points for which the Birkhoff average of φ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every β-shift satisfies almost specification and we show that the irregular set for any β-shift or β-transformation is either empty or has full topological entropy and Hausdorff dimension.