Tips of tongues in the double standard family*

Abstract

Abstract We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree two circle maps F λ : R / Z → R / Z defined by F λ ( x ) ≔ 2 x + a + b π sin ( 2 π x ) with λ ≔ ( a , b ) ∈ R / Z × ( 0 , 1 ) . We prove that if F λ ◦ n − i d has a zero of multiplicity three in R / Z , then there is a system of local coordinates ( α , β ) : W → R 2 defined in a neighborhood W of λ, such that α(λ) = β(λ) = 0 and F μ ◦ n − i d has a multiple zero with μ ∈ W if and only if β 3(μ) = α 2(μ). This shows that the tips of tongues are regular cusps.