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Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous mapwith the specification property, and $\varphi: X \mapsto \IR$ a continuousfunction. We consider the set of points for which the Birkhoff average of$\varphi$ does not exist (which we call the irregular set for $\varphi$) andshow that this set is either empty or carries full topological pressure (in thesense of Pesin and Pitskel). We formulate various equivalent natural conditionson $\varphi$ that completely describe when the latter situation holds and giveexamples of interesting systems to which our results apply but were notpreviously known. As an application, we show that for a suspension flow over acontinuous map with specification, the irregular set carries full topologicalentropy.

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