Tips of tongues in the double standard family* *The first author was supported by by FRPDF allotment 2018–19 of Presidency University. The second and third authors were supported by ANR grant Lambda ANR-13-BS01-0002. The third author was also supported by Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445), and

Abstract

We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree two circle maps defined by We prove that if has a zero of multiplicity three in , then there is a system of local coordinates defined in a neighborhood W of λ, such that α(λ) = β(λ) = 0 and 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.