Equilateral triangulations and the postcritical dynamics of meromorphic functions (Q6061988)

A result from a recent paper of the first author and \textit{L. Rempe} [``Non-compact Riemann surfaces are equilaterally triangulable'', Preprint, arXiv:2103.16702 [math.CV] (2021)] leads to the conclusion that for any domain \(D\subset\hat{\mathbb{C}}\) it is possible to find its equilateral triangulation. This means the existence of a countable and locally finite collection of closed topological triangles in \(D\), covering \(D\), such that any two triangles intersect only in a full edge or a vortex and for any two triangles sharing an edge there is an anti-conformal map of one triangle onto the other, fixing pointwise the common edge and transforming the remaining vertex of one triangle to the remaining vertex of the other. The authors take a step further and prove in Theorem B the existence of an equilateral triangulation with an additional condition on every triangle \(T\) containing a point \(z\): \[ \mbox{diameter}(T)\leq\eta(d(z,\partial D)), \] where the diameter is a spherical diameter and \(\eta:[0,\infty)\rightarrow[0,\infty)\) is continuous, strictly increasing with \(\eta(0)=0\). This condition means that any planar domain can be equilaterally triangulated with diameters of triangles tending to zero near the border of the domain. The existence of such a triangulation implies then the existence of a so-called Belyi function on \(D\), that is, a function \(g: D\rightarrow\hat{\mathbb{C}}\) branching only over \(\pm1\) and \(\infty.\) Further, appliying the Measureable Riemann Mapping Theorem and a refinement of a fixpoint technique introduced earlier by the first two authors [Math. Ann. 375, No. 3--4, 1761--1782 (2019; Zbl 1445.30013)], the authors proceed to prove the main result of the paper stating that a dynamics on a planar set \(S\) discrete in a domain \(D\) can be essentially realized by the postcritical dynamics of a function holomorphic in \(D.\) Theorem A. Let \(D\subseteq\hat{\mathbb{C}}\) be a domain, \(S\subset D\) a discrete set with \(|S|\geq3,\;h:S\rightarrow S\) a map, and \(\varepsilon>0\). Then there exists an \(\varepsilon\)-homeomorphism \(\phi:\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}\) and a holomorphic map \(f:\phi(D)\rightarrow\hat{\mathbb{C}}\) with no asymptotic values such that \(P(f)\subset\phi(D)\) and \(f|_{P(f)}:P(f)\rightarrow P(f)\) is \(\varepsilon\)-conjugate to \(h: S\rightarrow S\). Here \(P(f)\) denotes the postsingular set of \(f\), that is, \[ P(f)=\{f^n(w):\,w\in S(f),\,n\geq0\}, \] where \(S(f)\) is a set of singular (critical and asymptotic) values of \(f\). Also, the term \(\varepsilon\)-conjugacy refers to the situation, when a conjugacy \(\phi\) of two dynamical systems is an \(\varepsilon\)-homeomorphism, that is, fulfills the condition \(\sup\,d(\phi(z),z)<\varepsilon\).