Topological pressure on noncompact sets

Points with maximal Birkhoff average oscillation

“…Therefore, the set X ϕ,f is often ignored in ergodic theory. However, there is now an extensive literature showing that the irregular sets can be large from other points of view, such as from the points of view of topological entropy and Hausdorff dimension, see [4], [5], [8], [11], [13], [16], [21], [19], [17], [23], [26], [27], [32] and references therein. In particular, under the assumption that f satisfies the specification property, it was shown by Chen, Tassilo and Shu [8] that the irregular set X ϕ,f has full topological entropy and by Thompson [32] that it has full topological pressure if it is not empty.…”