The Irregular Set for Maps with the Specification Property Has Full Topological Pressure

Let (X, d) be a compact metric space, f : X 7→ X be a continuous map with the specification property, and φ : X 7→ R a continuous function. We consider the set of points for which the Birkhoff average of φ does not exist (which we call the irregular set for φ) and show that this set is either empty or carries full topological pressure (in the sense of Pesin and Pitskel). We formulate various equivalent natural conditions on φ that completely describe when the latter situation holds and give examples of interesting systems to which our results apply but were not previously known. As an application, we show that for a suspension flow over a continuous map with specification, the irregular set carries full topological entropy. For a compact metric space (X, d), a continuous map f : X 7→ X and a continuous function φ : X 7→ R, we define the irregular set for φ to be (1) X̂(φ) := { x ∈ X : lim n→∞ 1 n n−1 ∑ i=0 φ(f (x)) does not exist }