Counting conjugacy classes in groups with contracting elements

In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by (Formula presented.) (respectively, (Formula presented.)) the set of (respectively, primitive) conjugacy classes of algebraic length at most (Formula presented.) for a basepoint (Formula presented.). The main result is the following asymptotic formula: (Formula presented.) A similar formula holds for conjugacy classes using stable length. As a consequence of the formulae, the conjugacy growth series is transcendental for all non-elementary relatively hyperbolic groups, graphical small cancellation groups with finite components. As a by-product of the proof, we establish several useful properties for an exponentially generic set of elements. In particular, it yields a positive answer to a question of J. Maher that an exponentially generic element in mapping class groups has their Teichmüller axis contained in the principal stratum.