Irregular Sets for Piecewise Monotonic Maps

For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as linear mod $1$ transformations and generalized $\beta$-transformations), we show that the set of points for which the Birkhoff average of a continuous function does not exist (called the irregular set) is either empty or has full topological entropy. This generalizes Thompson's theorem for irregular sets of $\beta$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to Hofbauer-Raith problem on the density of periodic measures.