arXiv subjects

Roland Roeder

Publications and source records attributed to Roland Roeder.

14 recordsLinked to original sources

arXiv2026

Complex dynamics perspective for birational maps of the plane arising from cluster algebra mutations

Using the methods of holomorphic dynamics we investigate planar birational mappings that arise from the theory of cluster algebras and integrable systems. Computing dynamical degrees of these mappings, many of which are greater than one, allows us to show that many of the mappings do not have a conserved quantity (nor an invariant fibration). In most of the examples, invariant fibrations can also be ruled out by finding superattracting periodic points. This answers a question posted by Machacek and Ovenhouse 2024 and by Chen and Li 2024. Moreover, having found a good algebraically stable model for the mappings and having computed the dynamical degree, we can then apply results from the ergodic theory of birational maps to produce invariant measures with positive entropy and positive Lyapunov exponents.

math.DS↗

arXiv2025

A dynamical approach to studying the Lee-Yang zeros for the Potts Model on the Cayley Tree

Let $Z_n(z,t)$ denote the partition function of the $q$-state Potts Model on the rooted binary Cayley tree of depth~$n$. Here, $z = {\rm e}^{-h/T}$ and $t = {\rm e}^{-J/T}$ with $h$ denoting an externally applied magnetic field, $T$ the temperature, and $J$ a coupling constant. One can interpret $z$ as a ``magnetic field-like'' variable and $t$ as a ``temperature-like'' variable. Physical values $h \in \mathbb{R}$, $T > 0$, and $J \in \mathbb{R}$ correspond to $t \in (0,\infty)$ and $z \in (0,\infty)$. For any fixed $t_0 \in (0,\infty)$ and fixed $n \in \mathbb{N}$ we consider the complex zeros of $Z_n(z,t_0)$ and how they accumulate on the ray $(0,\infty)$ of physical values for $z$ as $n \rightarrow \infty$. In the ferromagnetic case ($J > 0$ or equivalently $t \in (0,1)$) these Lee-Yang zeros accumulate to at most one point on $(0,\infty)$ which we describe using explicit formulae. In the antiferromagnetic case $(J < 0$ or equivalently $t \in (1,\infty)$) these Lee-Yang zeros accumulate to finitely many points of $(0,\infty)$, which we again describe with explicit formulae. The same results hold for the unrooted Cayley tree of branching number two. These results are proved by adapting a renormalization procedure that was previously used in the case of the Ising model on the Cayley Tree by M\"uller-Hartmann and Zittartz (1974 and 1977), Barata and Marchetti (1997), and Barata and Goldbaum (2001). We then use methods from complex dynamics and, more specifically, the active/passive dichotomy for iteration of a marked point, along with detailed analysis of the renormalization mappings, to prove the main results.

math-ph↗

arXiv2025

Energy, equilibrium measure and entropy for toric surface maps

We consider the ergodic theory of plane rational maps that preserve the natural holomorphic volume form on the algebraic torus. Specifically we construct natural invariant probability measures for a large class of such maps by intersecting the equilibrium currents we constructed in our previous work [DR]. We show further that these measures are mixing and that each admits an underlying geometric product structure. The main result of [DDG3] then implies that the topological entropy of each map covered by our results is the log of its first dynamical degree. In light of examples presented in [BDJ], this implies in particular that the entropy of a rational map can equal the log of a transcendental number.

math.DS↗

arXiv2024

Equidistribution without stability for toric surface maps

We prove an equidistribution result for iterated preimages of curves by a large class of rational maps $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP}^2$ that cannot be birationally conjugated to algebraically stable maps. The maps, which include recent examples with transcendental first dynamical degree, are distinguished by the fact that they have constant Jacobian determinant relative to the natural holomorphic two form on the algebraic torus. Under the additional hypothesis that $f$ has "small topological degree'' we also prove an equidistribution result for iterated forward images of curves. To prove our results we systematically develop the idea of a positive closed $(1,1)$ current and its cohomology class on the inverse limit of all toric surfaces. This, in turn, relies upon a careful study of positive closed $(1,1)$ currents on individual toric surfaces. This framework may be useful in other contexts.

math.DS↗

arXiv2024

Questions about the dynamics on a natural family of affine cubic surfaces

We present several questions about the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces $$S_{A,B,C,D} = \{(x,y,z) \in \mathbb{C}^3 \, : \, x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D\},$$ where $A,B,C,$ and $D$ are complex parameters. This group action describes the monodromy of the famous Painlevé 6 Equation as well as the natural dynamics of the mapping class group on the ${\rm SL}(2,\mathbb{C})$ character varieties associated to the once punctured torus and the four times punctured sphere. The questions presented here arose while preparing our work ``Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé~6'' \cite{RR} and during informal discussions with many people. Several of the questions were posed at the Simons Symposium on Algebraic, Complex and Arithmetic Dynamics that was held at Schloss Elmau, Germany, in August 2022 as well as the MINT Summer School ``Facets of Complex Dynamics'' that was held in Toulouse, France, in June 2023.

math.AG↗

arXiv2024

Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6

We study the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces \begin{align*} S_{A,B,C,D} = \{(x,y,z) \in \mathbb{C}^3 \, : \, x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D\}, \end{align*} where $A,B,C,$ and $D$ are complex parameters. We focus on a finite index subgroup $Γ_{A,B,C,D} < {\rm Aut}(S_{A,B,C,D})$ whose action not only describes the dynamics of Painlevé 6 differential equations but also arises naturally in the context of character varieties. We define the Julia and Fatou sets of this group action and prove that there is a dense orbit in the Julia set. In order to show that the Julia set is ``large'' we consider a second dichotomy, between locally discrete and locally non-discrete dynamics. For an open set in parameter space, $\mathcal{N} \subset \mathbb{C}^4$, we show that there simultaneously exists an open set in $S_{A,B,C,D}$ on which $Γ_{A,B,C,D}$ acts locally discretely and a second open set in $S_{A,B,C,D}$ on which $Γ_{A,B,C,D}$ acts locally non-discretely. After removing a countable union of real-algebraic hypersurfaces from $\mathcal{N}$ we show that $Γ_{A,B,C,D}$ simultaneously exhibits a non-empty Fatou set and also a Julia set having non-trivial interior. The open set $\mathcal{N}$ contains a natural family of parameters previously studied by Dubrovin-Mazzocco. The interplay between the Fatou/Julia dichotomy and the locally discrete/non-discrete dichotomy plays a major theme in this paper and seems bound to play an important role in further dynamical studies of holomorphic automorphism groups.

math.DS↗

arXiv2022

Symmetries of the Three Gap Theorem

The Three Gap Theorem states that for any $α\in \mathbb{R}$ and $N \in \mathbb{N}$, the fractional parts of $\{ 0α, 1α, \dots, (N - 1)α\}$ partition the unit circle into gaps of at most three distinct lengths. We prove a result about symmetries in the order with which the sizes of gaps appear on the circle.

math.NT↗

arXiv2021

Level spacing statistics for the multi-dimensional quantum harmonic oscillator: algebraic case

We study the statistical properties of the spacings between neighboring energy levels for the multi-dimensional quantum harmonic oscillator that occur in a window $[E,E+ΔE)$ of fixed width $ΔE$ as $E$ tends to infinity. This regime provides a notable exception to the Berry-Tabor Conjecture from Quantum Chaos and, for that reason, it was studied extensively by Berry and Tabor in their seminal paper from 1977. We focus entirely on the case that the (ratios of) frequencies $ω_1,ω_2,\ldots,ω_d$ together with $1$ form a basis for an algebraic number field $Φ$ of degree $d+1$, allowing us to use tools from algebraic number theory. This special case was studied by Dyson, Bleher, Bleher-Homma-Ji-Roeder-Shen, and others. Under a suitable rescaling, we prove that the distribution of spacings behaves asymptotically quasiperiodically in $\log E$. We also prove that the distribution of ratios of neighboring spacings behaves asymptotically quasiperiodically in $\log E$. The same holds for the distribution of finite words in the finite alphabet of rescaled spacings. Mathematically, our work is a higher dimensional version of the Steinhaus Conjecture (Three Gap Theorem) involving the fractional parts of a linear form in more than one variable, and it is of independent interest from this perspective.

math.NT↗

arXiv2021

Chromatic Zeros On Hierarchical Lattices and Equidistribution on Parameter Space

Associated to any finite simple graph $Γ$ is the chromatic polynomial $P_Γ(q)$ whose complex zeroes are called the chromatic zeros of $Γ$. A hierarchical lattice is a sequence of finite simple graphs $\{Γ_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $μ_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $Γ_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $μ_n$ converges to some measure $μ$ as $n$ tends to infinity. We call $μ$ the limiting measure of chromatic zeros associated to $\{Γ_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $μ$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.

math-ph↗

arXiv2019

Pulling back singularities of codimension one objects

We prove that the preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map $g: (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^n,0)$ is again singular. This provides a generalization of previous results of this nature by Ebenfelt-Rothschild [Comm. Anal. Geom. 15 (2007), no. 2, 491-507], Lebl [arXiv:0812.2498], and Denkowski [Manuscripta Math. 149 (2016), no. 1-2, 83-91]. The same statement is proved for pullbacks of singular codimension one holomorphic foliations.

math.CV↗

arXiv2019

Problem on Mutant Pairs of Hyperbolic Polyhedra

We present a notion of mutation of hyperbolic polyhedra, analogous to mutation in knot theory, and then present a general question about commensurability of mutant pairs of polyhedra. We motivate that question with several concrete examples of mutant pairs for which commensurability is unknown. The polyhedra we consider are compact, so techniques involving cusps that are typically used to distinguishing mutant pairs of knots are not applicable. Indeed, new techniques may need to be developed to study commensurability of mutant pairs of polyhedra.

math.GT↗

arXiv2019

Lee-Yang-Fisher zeros for DHL and 2D rational dynamics, II. Global Pluripotential Interpretation

In a classical work of the 1950's, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee-Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee-Yang-Fisher (LYF) zeros. In this paper we continue studying their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the Migdal-Kadanoff renormalization transformation). We study properties of the Fatou and Julia sets of this transformation and then we prove that the Lee-Yang-Fisher zeros are equidistributed with respect to a dynamical (1,1)-current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also prove a more general equidistribution theorem which applies to rational mappings having indeterminate points, including the Migdal-Kadanoff renormalization transformation of various other hierarchical lattices.

math.DS↗

arXiv2016

Typical dynamics of plane rational maps with equal degrees

Let $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP^2}$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms $T:\mathbb{CP}^2\to\mathbb{CP}^2$, the perturbed map $T\circ f$ admits exactly two ergodic measures of maximal entropy $\log d$, one of saddle and one of repelling type. Neither measure is supported in an algebraic curve, and $T\circ f$ is `fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation. Absence of an invariant foliation extends to all $T$ outside a countable union of algebraic subsets. Finally, we illustrate all of our results in a more concrete particular instance connected with a two dimensional version of the well-known quadratic Chebyshev map.

math.DS↗

arXiv2010

Lee-Yang zeros for DHL and 2D rational dynamics, I. Foliation of the physical cylinder

In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising models always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $\RR$ in two variables (the renormalization transformation). We prove that $\RR$ is partially hyperbolic on an invariant cylinder $\CC$. The Lee-Yang zeros are organized in a transverse measure for the central-stable foliation of $\RR|\, \CC$. Their distribution is absolutely continuous. Its density is $C^\infty$ (and non-vanishing) below the critical temperature. Above the critical temperature, it is $C^\infty$ on a open dense subset, but it vanishes on the complementary Cantor set of positive measure. This seems to be the first occasion of a complete rigorous description of the Lee-Yang distributions beyond 1D models.

math.DS↗