Rigidity of analytic and smooth bi-cubic multicritical circle maps with bounded type rotation numbers

We prove that if two analytic multicritical circle maps with the same bounded type rotation number are topologically conjugate by a conjugacy which matches the critical points of the two maps while preserving the orders of their criticalities, then the conjugacy necessarily has $C^{1+\alpha }$ regularity, where $\alpha$ depends only on the bound on the type of the rotation number. We then extend this rigidity result to $C^3$-smooth bi-cubic circle maps.