On the dynamics of n-circle inversion

The article deals with singular perturbation of polynomial maps where is a complex parameter and n is the degree, which is a particular case of the family of rational maps known as McMullen maps. Our main result shows that even when the geometric limit of the Julia set converges to the unit circle or the annulus for a.e. Lebesgue , as n tends to infinity, the measure of maximal entropy always converges to the Lebesgue measure supported on the unit circle. Additionally we describe the dynamics on the Julia set and show that is related to a quotient of a shift of n symbols by an equivalence relation. Finally we prove that the Thurston pull-back map associated with a particular four-circle inversion is a ramified Galois covering. From the arithmetical point of view we prove that each n-circle inversion can be defined over its field of moduli.