Abstract
The nonarea-preserving spectrum of the renormalization operator for golden circles is computed at area-preserving fixed points in terms of the area-preserving spectrum. Under reasonable assumptions on the critical fixed point, it follows that the only essential unstable nonarea-preserving direction from the fixed point is the one already known with eigenvalue golden ratio, corresponding to perturbations with constant Jacobian not equal to 1. Consequences are drawn for families of nonarea-preserving maps with not necessarily constant Jacobian near 1. Furthermore, the result lends support to a larger renormalization picture which unites those for circle maps and area-preserving maps.