Related papers: Elastic Graphs for Main Molecule Matings

In this paper, we study CTP maps, that is, marked rational maps with constant Thurston pullback mapping. We prove that all the regular or mixing CTP polynomials satisfy McMullen's condition. Additionally, we construct a new class of…

We obtain an analogue of the prime number theorem for a class of branched covering maps on the $2$-sphere called expanding Thurston maps $f$, which are topological models of some rational maps without any smoothness or holomorphicity…

Every expanding Thurston map $f$ without periodic critical points is known to have an iterate $f^n$ which is the topological mating of two polynomials. This has been examined by Kameyama and Meyer; the latter who has offered an explicit…

Let {f_t} be any algebraic family of rational maps of a fixed degree, with a marked critical point c(t). We first prove that the hypersurfaces of parameters for which c(t) is periodic converge as a sequence of positive closed (1,1) currents…

We show the resolution of indeterminacy of rational maps from a regular surface to a tame stack locally of finite type over an excellent scheme. The proof uses the valuative criterion for proper tame morphisms, which was proved by Bresciani…

We prove that if $F$ is a degree $3$ Thurston map with two fixed critical points, then any irreducible obstruction for $F$ contains a Levy cycle. As a corollary, it will be shown that if $f$ and $g$ are two postcritically finite cubic…

We introduce the notion of n-mating in this work, which includes the classical mating of polynomials as a special case. The new notion brings further links between the polynomial world and the rational world than the classical one, as well…

In this article we provide a combinatorial sufficient (and conjecturally, necessary) condition (called $\alpha$-symmetry) for the mating of two postcritically finite polynomials in $\mathcal{S}_1$ to be obstructed. To do this, we study the…

We give a simple algorithm that determines whether a given post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, the algorithm produces the Hubbard tree; otherwise, the algorithm produces the…

We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and…

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.

A holomorphic endomorphism of $\mathbb{CP}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study…

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $\mu_\phi$ for each expanding Thurston map $f: S^2\rightarrow S^2$ together with a real-valued H\"older continuous potential $\phi$.…

The "openness" of a complex polynomial mapping is discussed and applied to the Fundamental Theorem of Algebra. In this category fall proofs of S. Wolfenstein, R.L. Thompson, J. Milnor, and S. Reich-S. Smale. These proofs take into account…

This note will describe an effective procedure for constructing critically finite real polynomial maps with specified combinatorics.

Nearly Euclidean Thurston (NET) maps are described by simple diagrams which admit a natural notion of size. Given a size bound $C$, there are finitely many diagrams of size at most $C$. Given a NET map $F$ presented by a diagram of size at…

We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian…

Let $f: S^2 \to S^2$ be a postcritically finite branched covering map without periodic branch points. We give necessary and sufficient algebraic conditions for $f$ to be homotopic, relative to its postcritical set, to an expanding map $g$.

Given a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps…

This paper constructs a combinatorial model for all postcritically finite rational maps arising as the Newton's method of a complex polynomial. This model is used in [LMS] to give a combinatorial classification of postcritically finite…