Abstract:

In this talk we will present the concept of textit{tongue} for families of degree 2 covering maps of the circle. This concept was introduced by Misiurewicz and Rodrigues in analogy to Arnold tongues of orientation preserving homeomorphisms of the circle.  Afterwards, we will analyse how these tongues extend beyond their natural range of definition, leading to the concept of textit{extended tongues}. Finally,  we will study  the properties of extended tongues for a family of degree 4 Blashcke products which restricts to degree 2 maps of the circle.