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To investigate the degree $d$ connectedness locus, Thurston studied \emph{$\sigma_d$-invariant laminations}, where $\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials $f_c(z) = z^2 +c$. In the same spirit, we consider the space of all \emph{cubic symmetric polynomials} $f_\lambda(z)=z^3+\lambda^2 z$ in three articles. In the first one we construct the lamination $C_sCL$ together with the induced factor space $\mathbb{S}/C_sCL$ of the unit circle $\mathbb{S}$. As will be verified in the third paper, $\mathbb{S}/C_sCL$ is a monotone model of the \emph{cubic symmetric connectedness locus}, i.e., the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for constructing $C_sCL$ analogous to the Lavaurs algorithm for constructing a combinatorial model $\mathcal{M}^{comb}_2$ of the Mandelbrot set $\mathcal{M}_2$., Comment: 27 pages, 3 figures. arXiv admin note: text overlap with arXiv:2201.11434

To investigate the degree $d$ connectedness locus, Thur\-ston studied \emph{$\sigma_d$-invariant laminations}, where $\sigma_d$ 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 \emph{cubic symmetric polynomials} $f_\lambda(z)=z^3+\lambda^2 z$ in a series of three articles. In the present paper, the first in the series, we construct a lamination $C_sCL$ together with the induced factor space ${\mathbb{S}}/C_sCL$ of the unit circle ${\mathbb{S}}$. As will be verified in the third paper of the series, ${\mathbb{S}}/C_sCL$ is a monotone model of the \emph{cubic symmetric connected locus}, i.e. the space of all cubic symmetric polynomials with connected Julia sets., Comment: 37 pages, 4 figures

We find a constructive bound for the word length of a generating set for the centralizer of an element of the Mapping Class Group. As a consequence, we show that it is algorithmically decidable whether two postcritically finite branched coverings of the sphere are Thurston equivalent., Comment: arXiv admin note: text overlap with arXiv:1310.1492

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., Comment: wrt version 1: Appendix B is revised -- the proof of Theorem B.8 is simplified

We study the group of self-equivalences of a partially postcritically finite branched cover and answer a question of Adam Epstein about contractibility of certain deformation spaces of rational maps., Comment: Section 7 on connection with postcritically finite maps and Appendix with classification of equalizing multicurves for Per'4 were added

The key result in the present paper is a direct analogue of the celebrated Thurston's Theorem 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 the two-sphere.

Associated to a Thurston map $f: S^2 \to S^2$ with postcritical set $P$ are several different invariants obtained via pullback: a relation on the set of free homotopy classes of curves in $S^2- P$, a linear operator on the free $\R$-module generated by these homotopy classes of curves, a virtual endomorphism on the pure mapping class group, an analytic self-map of an associated Teichmueller space, and an analytic self-correspondence on an associated moduli space. Viewing all of these objects as invariants of $f$, we investigate harmonious relationships between their properties., Comment: 48 pages, 3 figures

Let f be an obstructed Thurston map with canonical obstruction \Gamma_f. We prove the following generalization of Pilgrim's conjecture: if the first-return map F of a periodic component C of the topological surface obtained from the sphere by pinching the curves of \Gamma_f is a Thurston map then the canonical obstruction of F is empty. Using this result, we give a complete topological characterization of canonical Thurston obstructions., Comment: 28 pages

Let $f$ be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map $\sigma_f$ of a finite-dimensional Teichm\"uller space. We prove that this map extends continuously to the augmented Teichm\"uller 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 Teichm\"uller 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., Comment: revised version, 28 pages