Ergodic optimization for Gauss's continued fraction map

The theory of ergodic optimization for distance-expanding maps is extended to Gauss's continued fraction map. Since the set of invariant probability measures is not weak$^*$ closed, we establish a characterisation of the closure of this set, and investigate limit-maximizing measures for H\"older continuous functions. Although a Ma\~n\'e cohomology lemma is shown to hold, the typical periodic optimization conjecture is shown to fail, as a consequence of the typical finite optimization property established for a certain class of (rationally maximized) functions. The typical periodic optimization (TPO) property is shown to hold, however, for the class of $\alpha$-H\"older essentially compact functions.