Non-ergodicity of the geodesic flow on Cantor tree surfaces

Abstract: A Riemann surface equipped with its conformal hyperbolic metric is parabolic if and only if the geodesic flow on its unit tangent bundle is ergodic. Let X be a Cantor tree or a blooming Cantor tree Riemann surface. Fix a geodesic pants decomposition of X and call the boundary geodesics in the decomposition cuffs. Basmajian, Hakobyan, and \vSari\'c proved that if the lengths of cuffs are rapidly converging to zero, then X is parabolic. More recently, \vSari\'c proved a slightly slower convergence of lengths of cuffs to zero implies X is not parabolic. In the paper, we interpolate between the two rates of convergence of the cuffs to zero and find that these surfaces are not parabolic, thus completing the picture.