相关论文: Unmating of expanding Thurston maps with Julia set…

Let $f$ be a rational map with degree at least two. We prove that $f$ has at least $2$ disjoint and infinite critical orbits in the Julia set if it has a Herman ring. This result is sharp in the following sense: there exists a cubic…

According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia…

We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which…

Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have…

We realize a dynamical decomposition for a post-critically finite rational map which admits a combinatorial decomposition. We split the Riemann sphere into two completely invariant subsets. One is a subset of the Julia set consisting of…

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 post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm.…

Thurston introduced \emph{invariant (quadratic) laminations} in his 1984 preprint as a vehicle for understanding the connected Julia sets and the parameter space of quadratic polynomials. Important ingredients of his analysis of the angle…

We obtain an analog of the prime number theorem for a class of branched covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness…

In this paper, we consider random iterations of polynomial maps $z^2 +c_n$ where $c_n$ are complex-valued independent random variables following the uniform distribution on the closed disk with center $c$ and radius $r$. The aim of this…

In this paper we prove existence of matings between a large class of renormalizable cubic polynomials with one fixed critical point and another cubic polynomial having two fixed critical points. The resulting mating is a Newton map. Our…

Recent work of Dylan Thurston gives a condition for when a post-critically finite branched self-cover of the sphere is equivalent to a rational map. We apply D. Thurston's positive criterion for rationality to give a new proof of a theorem…

For every polynomial f of degree n with no double roots, there is an associated family C(f) of harmonic algebraic curves, fibred over the circle, with at most n-1 singular fibres. We study the combinatorial topology of C(f) in the generic…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

We investigate the family of marked Thurston maps that are defined everywhere on the topological sphere $S^2$, potentially excluding at most countable closed set of essential singularities. We show that when an unmarked Thurston map $f$ is…

Using Thurston's characterization of postcritically finite rational functions as branched coverings of the sphere to itself, we give a new method of constructing new conformal dynamical systems out of old ones. Let $f(z)$ be a rational map…

We study the asymptotic expansion of smooth one-dimensional maps. We give an example of an interval map for which the optimal shrinking of components exponential rate is not attained for any neighborhood of a certain fixed point in the…

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…

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative,…

The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…