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Daniel Meyer

Publications and source records attributed to Daniel Meyer.

18 recordsLinked to original sources
arXiv2026

Quasi-visual approximations

We develop the foundations of the theory of quasi-visual approximations of bounded metric spaces. Roughly speaking, these are sequences of covers of a given space for which the diameters of the sets in the covers shrink to zero and for which relative metric quantities (such as ratios of diameters and distances) are uniformly controlled. This framework has applications to questions in quasiconformal geometry. In particular, quasi-visual approximations can be used to detect whether a given homeomorphism between two bounded metric spaces is a quasisymmetry. We also explore the connection to the theory of Gromov hyperbolic spaces via the tile graph associated with a quasi-visual approximation. As an application, we relate these ideas to the dynamics of semi-hyperbolic rational maps. More specifically, we show that the Julia set of a rational map admits a dynamical quasi-visual approximation if and only if the map is semi-hyperbolic.

math.CV↗
arXiv2025

Extending Rational Expanding Thurston Maps

We consider postcritically finite rational maps $f\colon \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ whose Julia set is the whole Riemann sphere $\widehat{\mathbb{C}}$. We call such a map an expanding rational Thurston map. Identifying $\widehat{\mathbb{C}}$ with the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$, we show that $f$ may be extended on a neighborhood $\Omega\subset \mathbb{R}^3$ of $\widehat{\mathbb{C}}$ to a quasi-regular map $F\colon \Omega \to \mathbb{R}^3$. In fact, $F$ is uniformly quasi-regular in the following sense. The sequence of iterates $F^n$, each of which is defined on a neighborhood $\Omega_n$ of $\widehat{\mathbb{C}}= \mathbb{S}^2 \subset \mathbb{R}^3$, is uniformly quasi-regular. Here $\Omega_n$ shrink to $\widehat{\mathbb{C}}$, meaning that $\bigcap \Omega_n = \widehat{\mathbb{C}}$. This result may be viewed as a non-homeomorphic version of the extension of a quasi-conformal mapping $f:\mathbb{R}^2\to \mathbb{R}^2$ to a quasi-conformal mapping $F\colon \mathbb{R}^3 \to \mathbb{R}^3$ due to Ahlfors.

math.CV↗
arXiv2025

The Role of Legacy Mobile Networks in Infrastructure Resilience: Evidence from the Southern Brazil Flood

This paper investigates the resilience of mobile communication networks during the extreme flooding that affected Rio Grande do Sul, Brazil, in May 2024. Based on regulatory data and technical insights from operators, the study identifies the leading causes of mobile network disruptions, primarily related to flooding and prolonged power outages. The results reveal the significant vulnerability of modern networks (4G/5G) during the event and the essential role played by legacy technologies (2G/3G) in sustaining basic connectivity under adverse conditions. The findings underscore the necessity of disaster-aware infrastructure planning, taking into account the ongoing significance of legacy systems, diversified power supply strategies, and resilient network designs to enhance service continuity during future crises.

cs.NI↗
arXiv2023

Approximating Riemannian manifolds by polyhedra

This is a study on approximating a Riemannian manifold by polyhedra. Our scope is understanding Tullio Regge's [52] article in the restricted Riemannian frame. We give a proof of the Regge theorem along lines close to its original intuition: one can approximate a compact domain of a Riemannian manifold by polyhedra in such a way that the integral of the scalar curvature is approximated by a corresponding polyhedral curvature.

math.DG↗
arXiv2020

Quasiconformal and geodesic trees

A quasiconformal tree is a metric tree that is doubling and of bounded turning. We prove that every quasiconformal tree is quasisymmetrically equivalent to a geodesic tree with Hausdorff dimension arbitrarily close to 1.

math.MG↗
arXiv2020

Uniformly branching trees

A quasiconformal tree $T$ is a (compact) metric tree that is doubling and of bounded turning. We call $T$ trivalent if every branch point of $T$ has exactly three branches. If the set of branch points is uniformly relatively separated and uniformly relatively dense, we say that $T$ is uniformly branching. We prove that a metric space $T$ is quasisymmetrically equivalent to the continuum self-similar tree if and only if it is a trivalent quasiconformal tree that is uniformly branching. In particular, any two trees of this type are quasisymmetrically equivalent.

math.CV↗
arXiv2017

Expanding Thurston Maps

We study the dynamics of Thurston maps under iteration. These are branched covering maps $f$ of 2-spheres $S^2$ with a finite set $\mathop{post}(f)$ of postcritical points. We also assume that the maps are expanding in a suitable sense. Every expanding Thurston map $f\: S^2 \to S^2$ gives rise to a type of fractal geometry on the underlying sphere $S^2$. This geometry is represented by a class of \emph{visual metrics} $\varrho$ that are associated with the map. Many dynamical properties of the map are encoded in the geometry of the corresponding {\em visual sphere}, meaning $S^2$ equipped with a visual metric $\varrho$. For example, we will see that an expanding Thurston map is topologically conjugate to a rational map if and only if $(S^2, \varrho)$ is quasisymmetrically equivalent to the Riemann sphere $\widehat{\mathbf{C}}$. We also obtain existence and uniqueness results for $f$-invariant Jordan curves $\mathcal{C}\subset S^2$ containing the set $\mathop{post}(f)$. Furthermore, we obtain several characterizations of Lattès maps.

math.DS↗
arXiv2016

Exponential growth of some iterated monodromy groups

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 exponential growth. These groups arise from polynomials. We show exponential growth of the $\operatorname{IMG}$ of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpiński carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose $\operatorname{IMG}$ has exponential growth.

math.DS↗
arXiv2015

Invariant Jordan curves of Sierpiski carpet rational maps

In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpiński carpet, then there is an integer $n_0$, such that, for any $n\ge n_0$, there exists an $R^n$-invariant Jordan curve $Γ$ containing the postcritical set of $R$.

math.DS↗
arXiv2013

On The Notions of Mating

The different notions of matings of pairs of equal degree polynomials are introduced and are related to each other as well as known results on matings. The possible obstructions to matings are identified and related. Moreover the relations between the polynomials and their matings are discussed and proved. Finally holomorphic motion properties of slow-mating are proved.

math.DS↗
arXiv2013

Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is semi-conjugate to $z^d\colon S^1\to S^1$, where $d$ is equal to the degree of $F$. More precisely, for such an $F$ we construct a Peano curve $γ\colon S^1\to S^2$ (onto), such that $F\circ γ(z) = γ(z^d)$ (for all $z\in S^1$).

math.CV↗
arXiv2012

Expanding Thurston maps as quotients

A Thurston map is a branched covering map $f\colon S^2\to S^2$ that is postcritically finite. Mating of polynomials, introduced by Douady and Hubbard, is a method to geometrically combine the Julia sets of two polynomials (and their dynamics) to form a rational map. We show that for every expanding Thurston map $f$ every sufficiently high iterate $F=f^n$ is obtained as the mating of two polynomials. One obtains a concise description of $F$ via critical portraits. The proof is based on the construction of the invariant Peano curve from Meyer. As another consequence we obtain a large number of fractal tilings of the plane and the hyperbolic plane.

math.CV↗
arXiv2011

Unmating of rational maps, sufficient criteria and examples

Douady and Hubbard introduced the operation of mating of polynomials. This identifies two filled Julia sets and the dynamics on them via external rays. In many cases one obtains a rational map. Here the opposite question is tackled. Namely we ask when a given (postcritically finite) rational map $f$ arises as a mating. A sufficient condition when this is possible is given. If this condition is satisfied, we present a simple explicit algorithm to unmate the rational map. This means we decompose $f$ into polynomials, that when mated yield $f$. Several examples of unmatings are presented.

math.CV↗
arXiv2010

Quasicircles and Bounded Turning Circles Modulo bi-Lipschitz Maps

We construct a catalog, of snowflake type metric circles, that describes all metric quasicircles up to \bl\ equivalence. This is a metric space analog of a result due to Rohde. Our construction also works for all bounded turning metric circles; these need not be doubling. As a byproduct, we show that a metric quasicircle with Assouad dimension strictly less than two is bi-Lipschitz equivalent to a planar quasicircle.

math.CV↗
arXiv2010

Dimension of elliptic harmonic measure of Snowspheres

A metric space $\mathcal{S}$ is called a \defn{quasisphere} if there is a quasisymmetric homeomorphism $f\colon S^2\to \mathcal{S}$. We consider the elliptic harmonic measure, i.e., the push forward of 2-dimensional Lebesgue measure by $f$. It is shown that for certain self similar quasispheres $\mathcal{S}$ (snowspheres) the dimension of the elliptic harmonic measure is strictly less than the Hausdorff dimension of $\mathcal{S}$.

math.CV↗
arXiv2009

Snowballs are Quasiballs

We introduce snowballs, which are compact sets in $\R^3$ homeomorphic to the unit ball. They are 3-dimensional analogs of domains in the plane bounded by snowflake curves. For each snowball $B$ a quasiconformal map $f\colon \R^3\to \R^3$ is constructed that maps $B$ to the unit ball.

math.CV↗