  Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ 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 $\phi$ be a
continuous function on $X$. 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.
