Amalgamated Free Products of Circle Actions with a Bounded Number of Fixed Points

Inspired by constructions of Kova\v{c}evi\'{c}, we introduce the amalgamated free product of circle actions, obtained by blowing up two actions along prescribed orbits and rearranging the inserted intervals. Under natural orbit and index assumptions, we prove that this construction is well defined, yields a minimal action on the circle, and is unique up to topological conjugacy. We then study its dynamical properties. Using a proper ping-pong partition arising from the construction, we obtain criteria ensuring that the resulting action still has a uniformly bounded number of fixed points, and in particular at most $2n$ fixed points. We also give sufficient conditions for the resulting action to remain M\"obius-like and for it not to be topologically conjugate to a subgroup of any finite lift $\psl^{(k)}(2,\RR)$.