Higher-order fixed points of the renormalisation operator for invariant circles

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Abstract
A generalisation of the standard map with an additional term that has half the spatial period is studied. It is found that for certain parameter values this map lies on the stable manifold of a 3-cycle of the renormalisation operator that describes invariant circles with golden mean winding number. Examination of the behaviour of neighbouring maps under the renormalisation operator shows that this 3-cycle has at least two relevant unstable eigenvalues. Thus, there is at least a two-parameter family of maps that is invariant under the cube of the renormalisation operator. The significance of these structures arises from the fact that maps with invariant circles with golden mean winding number are attracted to a simple fixed point to the renormalisation operator.