Digital Object Identifier (DOI) 10.1007/s00220-003-0891-8 Commun. Math. Phys. 240, 75–96 (2003) Communications in Mathematical Physics Renormalization Horseshoe for Critical Circle Maps Michael Yampolsky ⋆ Mathematics Department, University of Toronto, 100 St. George Street, Toronto, Ontario, M5S 3G3, Canada Received: 3 October 2002 / Accepted: 7 March 2003 Published online: 10 July 2003 – © Springer-Verlag 2003 Abstract: We prove that the invariant horseshoe for the renormalization of critical cir- cle maps is globally uniformly hyperbolic with one-dimensional unstable direction. This extends our earlier result on hyperbolicity of periodic orbits of the renormalization, and completes the proof of Lanford’s Program. With this paper we complete our description of the renormalization picture for critical circle maps, begun in [Ya1, Ya2, Ya3 and EY]. The renormalization theory of critical circle maps serves to explain the universality phenomena which are observed in the families of one-dimensional dynamical systems given by smooth homeomorphisms of the circle with a finite number of (non-flat) critical points. Historically, this is one of the two main examples of universality in one-dimensional dynamics, the other being the Feigenbaum universality. A renormalization operator for critical circle maps appeared in the papers of Ostlund et al [ORSS] and Feigenbaum et al [FKS]. The most general formulation of the main conjectures of the theory is due to Lanford [Lan1, Lan2], and became known as Lanford’s Program. The renormalization theory of critical circle maps has developed in parallel with the other one-dimensional renormalization theory, that of the unimodal maps of the interval. The recent spectacular progress in the latter began with the seminal work of Sullivan [Sul1, Sul2, MvS], who gave a conceptual explanation of the existence of the Feigenbaum fixed point using the methods of holomorphic dynamics and the Teichm¨ uller theory. The results of Sullivan were adapted to the critical circle maps by de Faria [dF1, dF2], who, in particular, introduced an appropriate analogue of the Douady-Hubbard quadratic-like maps. McMullen’s approach to constructing the renormalization horseshoe for bounded type [McM2] has also been translated into the critical circle map setting by de Faria and de Melo [dFdM2]. In our works [Ya1,Ya2] we have extended the above mentioned results to all combinatorial types, and constructed the global renormalization horseshoe ⋆ The author gratefully acknowledges the financial support of NSERC and the Connaught Foundation of the University of Toronto.