We study a family of fractal sets, derived from Schottky groups, whose Hausdorff dimensions degenerate to zero. To compute the rate of this covergence, we employ the Selberg zeta function of the associated hyperbolic Schottky surfaces.
We prove that, after a suitable rescaling, these Selberg zeta functions converge to the Ihara zeta function of a limiting finite graph. This graph is associated to the relevant non-Archimedean Schottky group acting on the Berkovich projective line. From this, we also obtain the limiting behavior of resonances for the degenerating family of Schottky groups.
