Analytic results for the scaling behaviour of a piecewise-linear map of the circle

Abstract

For the piecewise-linear circle map theta to theta ', with theta ' identical to theta + Omega -K-(1/2- mod theta (mod 1)-1/2) the parameter values Omega i at which a periodic orbit, starting at theta =0 with winding number Fi/Fi+1, where Fi is the ith Fibonacci number, exists, are calculated analytically. These calculations are done at two K values, K=1, the critical case, and at K= kappa <1 (where ln(1- kappa )/ln(1+ kappa )=-(1+ square root 5)/2). At K= kappa the usual scaling behaviour for a smooth subcritical map is found, i.e. the same delta as Shenker (1982) found numerically. However, at K=1 a different critical delta value than is usually found numerically for smooth maps is calculated analytically for this piecewise-linear map