On Modular Smoothing and Scaling Functions for Mode Locking

The mode-locking structure of the sine circle map is investigated using the method of modular smoothing. It is shown that the method leads to a scaling function generated by the Gauss transformation. We speculate about a recursive procedure to obtain increasingly smooth descriptions of the fractal structure based on this method. T HE quasiperiodic transition to chaos in families of maps on the circle has been extensively studied in the past [1, 2, 3, 4, 5, 6]. It is known that the transition occurs at a parameter value at which the map ceases to be a diffeomorphism. Typically, this happens with the appearance of a cubic inflection point, and at greater values of the parameter, the map becomes non-invertible. A typical example of this class of maps is the sine circle map f( ) = + k 2 sin2 mod 1: (1) QMW preprint DYN #91-7, Phys. Lett. A, 163, 63–67, 1992 A CONICET (Consejo Nacional de Investigaciones Cientificas y Técnicas de Argentina) Fellow. Present Address: Institut Nonlinéaire de Nice, Université de Nice— Sophia Antipolis, Parc Valrose, 06034 Nice Cédex, France