RENORMALIZATION AND TRANSITION TO CHAOS IN AREA-PRESERVING NONTWIST MAPS

The problem of transition to chaos, i.e. the destruction of invariant circles or KAM (Kolmogorov-Arnold-Moser) curves, in area preserving no ntwist maps is studied within the renormalization group framework. Non twist maps are maps for which the twist condition is violated along a curve known as the shearless curve. In renormalization language this p roblem is that of finding and studying the fixed points of the renorma lization group operator R that acts on the space of maps. A simple per iod-two fixed point of R, whose basin of attraction contains the nontw ist maps for which the shearless curve exists, is found. Also, a criti cal period-12 fixed point of R, with two unstable eigenvalues, is foun d. The basin of attraction of this critical fixed point contains the n ontwist maps for which the shearless curve is at the threshold of dest ruction. This basin defines a new universality class for the transitio n to chaos in area preserving maps.