Fractal dimension associated with a critical circle map with an arbitrary-order inflection point

We have studied critical circle maps of the type f, with f =©+-2-z-1/2, -1/2<<1/2, and f(+1)=1+f, where z is the order of the inflection point of the angular variable. Such maps are believed to be useful in the study of the two-frequency quasiperiodic route to chaos. Using a variety of numerical approaches, we have calculated the fractal dimension D associated with such maps as a function of z. The different approaches yield consistent values for D and the completeness of the staircase has also been checked at each order. By comparing 1-D with the width of the mode-locked interval ©(0/1) (which may be analytically determined as a function of z for this class of maps), we have established that the ratio appears to be roughly independent of z with a value of 1/3. This leads to the empirical formula DS1-[(z-1)/3](1/z)z/ (z-1), which predicts for z=3 that DS0.872, in good agreement with previous precise direct numerical determinations. This general result suggests that the average width of mode-locked intervals of cycle length Q declines at the rate of Q-2/D with the 0/1 interval setting the scale for arbitrary intervals for all z and thereby governing the fractal dimension of the set complementary to the devils staircase.