Tips of Tongues in the Double Standard Family

2019/03/05 by Banerjee, Kuntal, Buff, Xavier, Canela, Jordi +1 · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.01795

Abstract

We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree 2 circle maps F_λ:ℝ/ℤ→ ℝ/ℤ defined by F_λ(x) := 2x + a+ \fracbπ sin(2πx)\quadwith λ:=(a,b)∈ ℝ/ℤ× (0,1). We prove that if F_λ^{∘ n}-\rm id has a zero of multiplicity 3 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 F_μ^{∘ n} - \rm 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.

Cited by

• Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets

Related

• Pinched Arnol'd tongues for Families of circle maps
• Boundaries of the Arnol'd tongues and the standard family
• Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets
• On the Piecewise Linear Perturbations of the Doubling Map
• Circles and Quadratic Maps Between Spheres