Genericity of pseudo-Anosov mapping classes, when seen as mapping classes

We prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes. More precisely, we consider a number of functions ρ on the mapping class group, and show that the proportion of pseudo-Anosov mapping classes with ρ-value at most R tends to 1 as R tends to infinity. The functions we consider include measuring the complexity of mapping classes using quasi-conformal distortion or Lipschitz distortion. We present a uniform approach to this problem using geodesic currents.