Limit $F$-signature functions of two-variable binomial hypersurfaces

The $F$-signature is a fundamental numerical invariant of singularities in positive characteristic. Its positivity detects strong $F$-regularity, an important class of singularities related to KLT singularities in characteristic zero. In this paper, we compute the limiting $F$-signature function of binomial and other related hypersurfaces in two variables as the characteristic $p\to \mathrm{\infty }$. In particular, we show it is a piecewise polynomial function, and relate it to the normalized volume.