  A general ansatz in Renormalization Theory, already established in many
important situations, states that exponential convergence of renormalization
orbits implies that topological conjugacies are actually smooth (when
restricted to the attractors of the original systems). In this paper we
establish this principle for a large class of bicritical circle maps, which are
$C^3$ circle homeomorphisms with irrational rotation number and exactly two
(non-flat) critical points. The proof presented here is an adaptation, to the
bicritical setting, of the one given by de Faria and de Melo for the case of a
single critical point. When combined with some recent papers, our main theorem
implies $C^{1+\alpha}$ rigidity for real-analytic bicritical circle maps with
rotation number of bounded type.
