Tongues in degree 4 Blaschke products (Q2834298)

The purpose of the paper is to describe the dynamical properties of the family of degree 4 Blaschke products \(B_{a}(z)=z^{3} \frac{z-a}{1-\overline{a} z}\), which is a rational family of perturbations of the doubling map. Firstly they study their basic topological properties and give some alternative parametrizations. They introduce the concept of tongues for the family \(B_{a}\) and focus on the tongue-like sets which appear in its parameter plane. The authors prove non-emptiness and connectivity of tongues and determine the boundary (Theorem A). The tongues are bounded for \(|a|<3\). If \(a\) belongs to the boundary of a tongue, then \(B_{a}\) has a parabolic cycle of multiplier 1. Further, the authors describe the bifurcations which take place in a neighborhood of the tips of the tongues (Theorem B). If the parameter is close enough to a tip, the two cycles leaving the unit circle are attracting. If \(|a|>2\) and \(B_{a}\) has a parabolic cycle on the unit circle, then \(a\) belongs to the boundary of a tongue. The authors describe also the shape of the connected components of the extended fixed tongue (Theorem C). Moreover, numerical studies suggest that the properties described for the extended fixed tongue are common to all extended tongues.