New fixed point of the renormalisation operator associated with the recurrence of invariant circles in generic Hamiltonian maps

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Abstract
The phenomenon of 'recurrent KAM tori' in twist maps is studied from a renormalisation point of view. A fixed point of the renormalisation operator for invariant circles, related to this phenomenon, is discussed. Connected to the fixed point are a three-cycle and a six-cycle. It is shown that for both the fixed point and the six-cycle there are homoclinic tangles giving rise to a basin of attraction of the simple fixed point with a very complicated shape. A piecewise-linear map is also studied. Even though the behaviour here seems related to the analytic case, it is found that in this case there is a fixed point, a family of two-cycles and a three-cycle for the renormalisation operator. The crossover behaviour is studied.