Let f be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map σ f of a finite-dimensional Teichmüller space. We prove that this map extends continuously to the augmented Teichmüller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston's pullback map near invariant strata of the boundary of the augmented Teichmüller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim and to infer further properties of Thurston's pullback map. Our approach also yields new proofs of Thurston's theorem and Pilgrim's Canonical Obstruction theorem.

To investigate the degree d connectedness locus, Thurston studied σ d -invariant laminations, where σ d is the d-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f (z) = z 2 + c. In the spirit of Thurston's work, we consider the space of all cubic symmetric polynomials f λ (z) = z 3 + λ 2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C s CL together with the induced factor space S/C s CL of the unit circle S. As will be verified in the third paper of the series, S/C s CL is a monotone model of the cubic symmetric connected locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

Associated to a Thurston map f : S 2 → S 2 with postcritical set P are several different invariants obtained via pullback: a relation S P f ←− S P on the set S P of free homotopy classes of curves in S 2 \ P , a linear operator λ f : R[S P ] → R[S P ] on the free R-module generated by S P , a virtual endomorphism φ f : PMod(S 2 , P ) PMod(S 2 , P ) on the pure mapping class group, an analytic self-map σ f : T (S 2 , P ) → T (S 2 , P ) of an associated Teichmüller space, and an analytic self-correspondence X • Y −1 : M(S 2 , P ) ⇒ M(S 2 , P ) of an associated moduli space. Viewing these associated maps as invariants of f , we investigate relationships between their properties.

In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of "Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. Then we prove that these periodic points are hyperbolic with one-dimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters. Contents 1. Introduction 1 2. Pacman renormalization operator 8 3. Siegel pacmen 15 4. Control of pullbacks 26 5. Maximal prepacmen 41 6. Maximal parabolic prepacmen 45 7. Hyperbolicity Theorem 55 8. Scaling Theorem 57 Appendix A. Sector renormalizations of a rotation 60 Appendix B. Lifting of curves under anti-renormalization 63 Appendix C. The molecule renormalization 81 References 85

The key result in the present paper is a direct analogue of the celebrated Thurston's Theorem Douady and Hubbard (Acta Math 171:263-297, 1993) for marked Thurston maps with parabolic orbifolds. Combining this result with previously developed techniques, we prove that every Thurston map can be constructively geometrized in a canonical fashion. As a consequence, we give a partial resolution of the general problem of decidability of Thurston equivalence of two postcritically finite branched covers of S 2 (cf. Bonnot et al. Moscow Math J 12:747-763, 2012).