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Roland K. W. Roeder

Publications and source records attributed to Roland K. W. Roeder.

At least 19 recordsLinked to original sources
arXiv2019

$q$-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs

We report exact results concerning the zeros of the partition function of the Potts model in the complex $q$ plane, as a function of a temperature-like Boltzmann variable $v$, for the $m$'th iterate graphs $D_m$ of the Diamond Hierarchical Lattice (DHL), including the limit $m \to \infty$. In this limit we denote the continuous accumulation locus of zeros in the $q$ planes at fixed $v = v_0$ as ${\mathcal B}_q(v_0)$. We apply theorems from complex dynamics to establish properties of ${\mathcal B}_q(v_0)$. For $v=-1$ (the zero-temperature Potts antiferromagnet, or equivalently, chromatic polynomial), we prove that ${\mathcal B}_q(-1)$ crosses the real-$q$ axis at (i) a minimal point $q=0$, (ii) a maximal point $q=3$ (iii) $q=32/27$, (iv) a cubic root that we give, with the value $q = q_1 = 1.6388969..$, and (v) an infinite number of points smaller than $q_1$, converging to $32/27$ from above. Similar results hold for ${\mathcal B}_q(v_0)$ for any $-1 < v < 0$ (Potts antiferromagnet at nonzero temperature). The locus ${\mathcal B}_q(v_0)$ crosses the real-$q$ axis at only two points for any $v > 0$ (Potts ferromagnet). We also provide computer-generated plots of ${\mathcal B}_q(v_0)$ at various values of $v_0$ in both the antiferromagnetic and ferromagnetic regimes and compare them to numerically computed zeros of $Z(D_4,q,v_0)$.

math-ph↗
arXiv2019

Limiting Measure of Lee--Yang Zeros for the Cayley Tree

This paper is devoted to an in-depth study of the limiting measure of Lee--Yang zeroes for the Ising Model on the Cayley Tree. We build on previous works of Müller-Hartmann-Zittartz (1974 and 1977), Barata--Marchetti (1997), and Barata--Goldbaum (2001), to determine the support of the limiting measure, prove that the limiting measure is not absolutely continuous with respect to Lebesgue measure, and determine the pointwise dimension of the measure at Lebesgue a.e. point on the unit circle and every temperature. The latter is related to the critical exponents for the phase transitions in the model as one crosses the unit circle at Lebesgue a.e. point, providing a global version of the "phase transition of continuous order" discovered by Müller-Hartmann-Zittartz. The key techniques are from dynamical systems because there is an explicit formula for the Lee-Yang zeros of the finite Cayley Tree of level $n$ in terms of the $n$-th iterate of an expanding Blaschke Product. A subtlety arises because the conjugacies between Blaschke Products at different parameter values are not absolutely continuous.

math-ph↗
arXiv2015

Around the boundary of complex dynamics

We introduce the exciting field of complex dynamics at an undergraduate level while reviewing, reinforcing, and extending the ideas learned in an typical first course on complex analysis. Julia sets and the famous Mandelbrot set will be introduced and interesting properties of their boundaries will be described. We will conclude with a discussion of problems at the boundary between complex dynamics and other areas, including a nice application of the material we have learned to a problem in astrophysics.

math.DS↗
arXiv2015

Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials

We study the two-point correlation functions for the zeroes of systems of $SO(n+1)$-invariant Gaussian random polynomials on $\mathbb{RP}^n$ and systems of ${\rm isom}(\mathbb{R}^n)$-invariant Gaussian analytic functions. Our result reflects the same "repelling," "neutral," and "attracting" short-distance asymptotic behavior, depending on the dimension, as was discovered in the complex case by Bleher, Shiffman, and Zelditch. For systems of the ${\rm isom}(\mathbb{R}^n)$-invariant Gaussian analytic functions we also obtain a fast decay of correlations at long distances. We then prove that the correlation function for the ${\rm isom}(\mathbb{R}^n)$-invariant Gaussian analytic functions is "universal," describing the scaling limit of the correlation function for the restriction of systems of the $SO(k+1)$-invariant Gaussian random polynomials to any $n$-dimensional $C^2$ submanifold $M \subset \mathbb{RP}^k$. This provides a real counterpart to the universality results that were proved in the complex case by Bleher, Shiffman, and Zelditch. (Our techniques also apply to the complex case, proving a special case of the universality results of Bleher, Shiffman, and Zelditch.)

math-ph↗
arXiv2015

Computing dynamical degrees

The dynamical degrees of a rational map $f:X\dashrightarrow X$ are fundamental invariants describing the rate of growth of the action of iterates of $f$ on the cohomology of $X$. When $f$ has nonempty indeterminacy set, these quantities can be very difficult to determine. We study rational maps $f:X^N\dashrightarrow X^N$, where $X^N$ is isomorphic to the Deligne-Mumford compactification $\overline {\mathcal M}_{0,N+3}$. We exploit the stratified structure of $X^N$ to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this article, we explicitly compute all of the dynamical degrees for all such maps $f:X^N\dashrightarrow X^N$, where $\mathrm{dim}(X^N)\leq 3$ and the first dynamical degrees for the mappings where $\mathrm{dim}(X^N)\leq 5$. These examples naturally arise in the setting of Thurston's topological characterization of rational maps.

math.DS↗
arXiv2014

The action on cohomology by compositions of rational maps

We use intuitive results from algebraic topology and intersection theory to clarify the pullback action on cohomology by compositions of rational maps. We use these techniques to prove a simple sufficient criterion for functoriality of a composition of two rational maps on all degrees of cohomology and we then reprove the criteria of Diller-Favre, Bedford-Kim, and Dinh-Sibony. We conclude with a cautionary example.

math.DS↗
arXiv2013

Superstable manifolds of invariant circles and co-dimension 1 Bottcher functions

We consider the situation of a dominant meromorphic self-map $f: X -rightarrow X$, where $X$ is a compact Kähler manifold of dimension $n > 1$. Suppose there is an embedded copy of $\mathbb{P}^1$ that is invariant under $f$, with $f$ holomorphic and transversally superattracting with degree $a$ in some neighborhood. Suppose $f$ restricted to this line is given by $z\mapsto z^b$, with resulting invariant circle $S$. We prove that if $a \geq b$, then the local stable manifold $W^s_\loc(S)$ is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition $a \geq b$ cannot be relaxed without adding additional hypotheses by presenting two examples with $a < b$ for which $W^s_\loc(S)$ is not real analytic in the neighborhood of any point.

math.DS↗
arXiv2012

Counting Zeros of Harmonic Rational Functions and Its Application to Gravitational Lensing

General Relativity gives that finitely many point masses between an observer and a light source create many images of the light source. Positions of these images are solutions of $r(z)=\bar{z},$ where $r(z)$ is a rational function. We study the number of solutions to $p(z) = \bar{z}$ and $r(z) = \bar{z},$ where $p(z)$ and $r(z)$ are polynomials and rational functions, respectively. Upper and lower bounds were previously obtained by Khavinson-Świcatek, Khavinson-Neumann, and Petters. Between these bounds, we show that any number of simple zeros allowed by the Argument Principle occurs and nothing else occurs, off of a proper real algebraic set. If $r(z) = \bar{z}$ describes an $n$-point gravitational lens, we determine the possible numbers of generic images.

math.CV↗
arXiv2011

Nearest Neighbor Distances on a Circle: Multidimensional Case

We study the distances, called spacings, between pairs of neighboring energy levels for the quantum harmonic oscillator. Specifically, we consider all energy levels falling between E and E+1, and study how the spacings between these levels change for various choices of E, particularly when E goes to infinity. Primarily, we study the case in which the spring constant is a badly approximable vector. We first give the proof by Boshernitzan-Dyson that the number of distinct spacings has a uniform bound independent of E. Then, if the spring constant has components forming a basis of an algebraic number field, we show that, when normalized up to a unit, the spacings are from a finite set. Moreover, in the specific case that the field has one fundamental unit, the probability distribution of these spacings behaves quasiperiodically in log E. We conclude by studying the spacings in the case that the spring constant is not badly approximable, providing examples for which the number of distinct spacings is unbounded.

math-ph↗
arXiv2011

A dichotomy for Fatou components of polynomial skew products

We consider polynomial maps of the form f(z,w) = (p(z),q(z,w)) that extend as holomorphic maps of CP^2. Mattias Jonsson introduces in (Math. Ann., 1999) a notion of connectedness for such polynomial skew products that is analogous to connectivity for the Julia set of a polynomial map in one-variable. We prove the following dichotomy: if f is an Axiom-A polynomial skew product, and f is connected, then every Fatou component of f is homeomorphic to an open ball; otherwise, some Fatou component of f has infinitely generated first homology.

math.DS↗
arXiv2010

Topology of Fatou components for endomorphisms of CP^k: Linking with the Green's Current

Little is known about the global topology of the Fatou set $U(f)$ for holomorphic endomorphisms $f: \mathbb{CP}^k \to \mathbb{CP}^k$, when $k >1$. Classical theory describes $U(f)$ as the complement in $ \mathbb{CP}^k$ of the support of a dynamically-defined closed positive $(1,1)$ current. Given any closed positive $(1,1)$ current $S$ on $ \mathbb{CP}^k$, we give a definition of linking number between closed loops in $\mathbb{CP}^k \setminus \supp S$ and the current $S$. It has the property that if $lk(γ,S) \neq 0$, then $γ$ represents a non-trivial homology element in $H_1(\mathbb{CP}^k \setminus \supp S)$. As an application, we use these linking numbers to establish that many classes of endomorphisms of $\mathbb{CP}^2$ have Fatou components with infinitely generated first homology. For example, we prove that the Fatou set has infinitely generated first homology for any polynomial endomorphism of $\mathbb{CP}^2$ for which the restriction to the line at infinity is hyperbolic and has disconnected Julia set. In addition we show that a polynomial skew product of $\mathbb{CP}^2$ has Fatou set with infinitely generated first homology if some vertical Julia set is disconnected. We then conclude with a section of concrete examples and questions for further study.

math.DS↗
arXiv2007

Computing arithmetic invariants for hyperbolic reflection groups

We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such groups and the isomorphism class of the invariant quaternion algebra by analyzing its ramification. We present many computed examples of these invariants. This is enough to show that most of the groups that we consider are pairwise incommensurable. For pairs of groups with identical invariants, not all is lost: when both groups are arithmetic, having identical invariants guarantees commensurability. We discover many ``unexpected'' commensurable pairs this way. We also present a non-arithmetic pair with identical invariants for which we cannot determine commensurability.

math.GT↗
arXiv2006

Constructing hyperbolic polyhedra using Newton's Method

We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's Method. Under the restriction that the dihedral angles are non-obtuse, Andreev's Theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyhedron realizing a given combinatorial structure $C$. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Our construction uses Newton's method and a homotopy to explicitly follow the existence proof presented by Andreev, providing both a very clear illustration of proof of Andreev's Theorem as well as a convenient way to construct three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. As an application, we construct compact hyperbolic polyhedra having dihedral angles that are (proper) integer sub-multiples of $π$, so that the group $Γ$ generated by reflections in the faces is a discrete group of isometries of hyperbolic space. The quotient $\mathbb{H}^3/Γ$ is hence a compact hyperbolic 3-orbifold, of which we study the hyperbolic volume and spectrum of closed geodesic lengths using SnapPea. One consequence is a volume estimate for a ``hyperelliptic'' manifold considered by Mednykh and Vesnin (see references).

math.GT↗
arXiv2006

Andreev's Theorem on hyperbolic polyhedra

In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing $C$ with the assigned dihedral angles. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Andreev's Theorem is both an interesting statement about the geometry of hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof for Thurston's Hyperbolization Theorem for 3-dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev's Theorem resembles (in a simpler way) the proof of Thurston. We correct a fundamental error in Andreev's proof of existence and also provide a readable new proof of the other parts of the proof of Andreev's Theorem, because Andreev's paper has the reputation of being ``unreadable''.

math.GT↗
arXiv2006

A degenerate Newton's Map in two complex variables: linking with currents

Little is known about the global structure of the basins of attraction of Newton's method in two or more complex variables. We make the first steps by focusing on the specific Newton mapping to solve for the common roots of $P(x,y) = x(1-x)$ and $Q(x,y) = y^2+Bxy-y$. There are invariant circles $S_0$ and $S_1$ within the lines $x=0$ and $x=1$ which are superattracting in the $x$-direction and hyperbolically repelling within the vertical line. We show that $S_0$ and $S_1$ have local super-stable manifolds, which when pulled back under iterates of $N$ form global super-stable spaces $W_0$ and $W_1$. By blowing-up the points of indeterminacy $p$ and $q$ of $N$ and all of their inverse images under $N$ we prove that $W_0$ and $W_1$ are real-analytic varieties. We define linking between closed 1-cycles in $W_i$ ($i=0,1$) and an appropriate positive closed $(1,1)$ current providing a homomorphism $lk:H_1(W_i,\mathbb{Z}) \to \mathbb{Q}$. If $W_i$ intersects the critical value locus of $N$, this homomorphism has dense image, proving that $H_1(W_i,\mathbb{Z})$ is infinitely generated. Using the Mayer-Vietoris exact sequence and an algebraic trick, we show that the same is true for the closures of the basins of the roots $\bar{W(r_i)}$.

math.DS↗