Tips of tongues in the double standard family

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 + πb sin(2πx) with λ:= (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 β3(μ) = α2(μ). This shows that the tips of tongues are regular cusps. © 2021 IOP Publishing Ltd & London Mathematical Society.