Pinched Arnol'd tongues for Families of circle maps

Abstract: The family of circle maps \begin{equation*} f_{b, \omega} (x) = x + \omega + b\, \phi(x) \end{equation*} is used as a simple model for a periodically forced oscillator. The parameter $\omega$ represents the unforced frequency, $b$ the coupling, and $\phi$ the forcing. When $\phi = \frac{1}{2 \pi} \sin(2 \pi x)$ this is the classical Arnol'd standard family. Such families are often studied in the $(\omega,b)$-plane via the so-called tongues $T_\beta$ consisting of all $(\omega,b)$ such that $f_{b, \omega}$ has rotation number $\beta$. The interior of the rational tongues $T_{p/q}$ represent the system mode-locked into a $p/q$-periodic response. Campbell, Galeeva, Tresser, and Uherka proved that when the forcing is a PL map with $k=2$ breakpoints, all $T_{p/q}$ pinch down to a width of a single point at multple values when $q$ large enough. In contrast, we prove that it generic amongst PL forcings with a given $k\geq 3$ breakpoints that there is no such pinching of any of the rational tongues. We also prove that the absence of pinching is generic for Lipschitz and $C^r$ ($r>0$) forcing.