LAUSR

We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree two circle maps Fλ:R/Z→R/Z defined by Fλ(x)≔2x+a+bπsin(2πx)withλ≔(a,b)∈R/Z×(0,1). We prove that if Fλ◦n−id has a zero of multiplicity three in R/Z , then there is a system of local coordinates (α,β):W→R2 defined in a neighborhood W of λ, such that α(λ) = β(λ) = 0 and Fμ◦n−id 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.