Ergodic Theory and Dynamical Systems

Complex bounds for renormalization of critical circle maps 1

a1 Mathematics Department, Yale University, New Haven, CT 06520-8283, USA (e-mail:


We use the methods developed with Lyubich for proving complex bounds for real quadratics to extend de Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for renormalizations of critical circle maps follows.

As another application of our methods we present a new proof of a theorem of Petersen on local connectivity of some Siegel Julia sets.

(Received May 4 1996)
(Revised December 12 1997)


1 This is an expanded version of the paper ‘Complex bounds for critical circle maps’, which appeared as IMS Stony Brook Preprint 95-12.