Complex bounds for critical circle maps

TL;DR

This paper extends complex bounds to all critical circle maps with irrational rotation numbers, using methods from quadratic maps, and applies these bounds to prove local connectivity of certain Julia sets.

Contribution

It generalizes complex bounds from quadratic maps to all critical circle maps with irrational rotation numbers, enabling new results in dynamical systems.

Findings

Contracting property for renormalizations established

Complex bounds extended to all critical circle maps with irrational rotation

Application to local connectivity of Julia sets

Abstract

We use the methods developed with M. Lyubich for proving complex bounds for real quadratics to extend E. 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. In the Appendix we give an application of the complex bounds for proving local connectivity of some Julia sets.