Related papers: A classification of postcritically finite Newton m…
A completely stable multicurve of a post-critically finite rational map induces a combinatorial decomposition. The projections of the small Julia sets are immersed within the original Julia set. We prove that two small Julia sets are…
For a rational function $R$, let $N_R(z)=z-\frac{R(z)}{R'(z)}.$ Any such $N_R$ is referred to as a Newton map. We determine all the rational functions $R$ for which $N_R$ has exactly two attracting fixed points, one of which is an…
The paper deals with Newton maps of complex exponential functions and a surgery tool developed by P. Ha\"issinsky. The concept of "Postcritically minimal" Newton maps of complex exponential functions are introduced, analogous to…
In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing…
A Thurston map $f\colon (S^2, A) \to (S^2, A)$ with marking set $A$ induces a pullback relation on isotopy classes of Jordan curves in $(S^2, A)$. If every curve lands in a finite list of possible curve classes after iterating this pullback…
In this paper, we study the dynamical uniform boundedness principle over a family of rational maps with certain nontrivial automorphisms. Specifically, we consider a family of rational maps of an arbitrary degree $d\ge 2$ whose automorphism…
In this paper, we define the core entropy for postcritically-finite Newton maps and study its continuity within this family. We show that the entropy function is not continuous in this family, which is different from the polynomial case…
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…
To approximate a simple root of an equation we construct families of iterative maps of higher order of convergence. These maps are based on model functions which can be written as an inner product. The main family of maps discussed is…
A study of real quadratic maps with real critical points, emphasizing the effective construction of critically finite maps with specified combinatorics. We discuss the behavior of the Thurston algorithm in obstructed cases, and in one…
A rational map $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ on the Riemann sphere $\widehat{\mathbb{C}}$ is called critically fixed if each critical point of $f$ is fixed under $f$. In this article, we study the properties of a…
In this paper, we consider a one-parameter family of degree $d\ge 2$ rational maps with an automorphism group containing the cyclic group of order $d$. We construct a polynomial whose roots correspond to parameter values for which 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…
The behavior under iteration of the critical points of polynomial maps plays an essential role in understanding its dynamics. We study the special case where the forward orbits of the critical points are finite. Thurston's theorem tells us…
An orientation-preserving branched covering map $f\colon S^2 \to S^2$ is called a critically fixed Thurston map if $f$ fixes each of its critical points. It was recently shown that there is an explicit one-to-one correspondence between…
This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large…
Mating is an operation to construct a rational map f from two polynomials, which are not in conjugate limbs of the Mandelbrot set. When the Thurston Algorithm for the unmodified formal mating is iterated in the case of postcritical…
We discuss the dynamical, topological, and algebraic classification of rational maps $f$ of the Riemann sphere to itself each of whose critical points $c$ is also a fixed-point of $f$, i.e. $f(c)=c$.
The generic linear evolution of the density matrix of a system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear…
The holomorphic endomorphism f of projective space is called post-critically finite (PCF) if the forward image of the critical locus, under iteration of f, has algebraic support (i.e., is a finite union of hypersurfaces). In the case of…