Abstract

Misieurewicz and Rodrigues introduced the the concept of tongues of the double standard family as open connected sets of parameters for which the double standard map has an attracting cycle (of a prescribed type). The boundary of a tongue consists of two curves which intersect tangentially on the tip of the tongue. Misieurewicz and Rodrigues described the order of tangency of these curves for a given tongue (called the fixed tongue) and conjectured that this order is general for all tongues of the family. In this talk we will show that this is true by proving that all tongues of the family form regular cusps. The proof relies on the transversality techniques introduced by Adam Epstein.