Related papers: Extending Rational Expanding Thurston Maps

Expanding Thurston maps form a class of branched covering maps on the topological $2$-sphere $S^{2}$, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumption initially…

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 the second paper [LZ24b] of this series, we obtained 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…

The set \[ \overline{\mathbb{E}}= \{ x \in {\mathbb{C}}^3: \quad 1-x_1 z - x_2 w + x_3 zw \neq 0 \mbox{ whenever } |z| < 1, |w| < 1 \} \] is called the tetrablock and has intriguing complex-geometric properties. It is polynomially convex,…

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 $\phi$.…

Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map \sigma_f of the Teichmueller space T modelled on P_f to itself fixing the basepoint corresponding to the identity map…

In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a…

We prove that if the Ahlfors regular conformal dimension $Q$ of a topologically cxc map on the sphere $f: S^2 \to S^2$ is realized by some metric $d$ on $S^2$, then either Q=2 and $f$ is topologically conjugate to a semihyperbolic rational…

Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as "Levy-free" maps and as…

The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to…

Let $M$ be a connected real-analytic hypersurface in $\C^N$ and $\S$ the unit real sphere in $\C^{N'}$, $N'> N\geq 2$. Assume that $M$ does not contain any complex-analytic hypersurface of $\C^N$ and that there exists at least one strongly…

A direct application of Zorn's Lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$ has an extension to a Lipschitz map $\widetilde f: \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$. This is analogous, but more easy,…

Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia iteration theory of rational functions extend to quasiregular self-maps of the Riemann sphere for which the degree exceeds the dilatation. We show that in this context,…

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…

The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire…

Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$-sphere $S^2$ induces a pull-back operation on Jordan curves $\alpha\subset S^2\setminus P_f$, where $P_f$ is the postcritical set of $f$. Here the isotopy class $[f^{-1}(\alpha)]$…

We show that in the family of degree $d\geq 2$ rational maps of the Riemann sphere, the closure of strictly postcritically finite maps contains a (relatively) Baire generic subset of maps displaying maximal non-statistical behavior: for a…

We develop the theory of Thurston maps that are defined everywhere on the topological sphere $S^2$ with a possible exception of a single essential singularity. We establish an analog of the celebrated W. Thurston's characterization theorem…

Let $S$ be a closed Riemann surface of genus $g(\geqq 2)$ and set $\dot{S}=S \setminus \{\hat{z}_0 \}$. Then we have the composed map $\varphi\circ r$ of a map $r: T(S) \times U \rightarrow F(S)$ and the Bers isomorphism $\varphi: F(S)…

We prove that every postsingularly finite entire map $g$ can be approximated by a sequence of postcritically finite complex polynomials $(g_n)$ such that their postsingular dynamics $g|P_g$ and $g_n|P_{g_n}$ are conjugate for every $n \in…