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Han Peters

Publications and source records attributed to Han Peters.

At least 19 recordsLinked to original sources
arXiv2026

Equidistribution Measures of infinite entropy for Transcendental Functions

In the 1980s Lyubich and Freire-Lopes-Ma\~n\'e proved that for any rational function of degree d \geq 2, both preimages and periodic points equidistribute to the unique measure of maximal entropy log(d). Their results provide a fundamental understanding of the dynamics of iterated rational functions, and have since been generalized to many different contexts, including classes of higher-dimensional polynomial and rational maps. In the current paper we depart from the algebraic category and aim to prove analogous statements for transcendental functions in the complex plane, which have infinite topological entropy. We introduce two different methods for constructing invariant measures in the transcendental setting, namely via embedded symbolic dynamical systems and via transfer operators associated to suitably chosen weights. In the latter case we isolate three properties of the weights -normality, tightness, and irreducibility- which together imply convergence to an invariant measure. We provide examples for each method, given by three classes of transcendental entire functions: disjoint-type maps, strongly polynomial-like maps, and a class of maps inspired by Baker's construction of multiply connected wandering domains and by Bishop's construction of Julia sets of Hausdorff dimension 1, which we call Baker-Bishop maps. For each of these classes we prove that with respect to carefully chosen weights, preimages equidistribute to an invariant mixing probability measure of infinite entropy. For Baker-Bishop maps and disjoint-type maps we also prove equidistribution of periodic points. In contrast to the rational setting, the measures we construct are not unique: by varying the weights one obtains infinitely many distinct measures.

math.DS↗
arXiv2026

Decay of correlations and zeros for the hard-core model

In a recent paper the last author proved that absence of complex zeros of the partition function of the hard-core model near a parameter $\lambda>0$ implies a form of correlation decay called strong spacial mixing. In this paper we investigate the reverse implication. We introduce a strengthening of strong spatial mixing that we call very strong spatial mixing (VSSM). Our main result is that if VSSM holds at a parameter $\lambda>0$ for a family of graphs, this implies that the partition function has no zeros near that parameter for each graph in the family. We also demonstrate that a closely related variant of very strong spatial mixing does not imply zero-freeness. As a consequence of our main result, we moreover obtain that VSSM implies spectral independence. Our proof relies on transforming the problem to the analysis of an induced non-autonomous dynamical system given by M\"obius transformations.

math.PR↗
arXiv2024

The independence polynomial on recursive sequences of graphs

We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are uniformly bounded. Each of the recursion algorithms leads to a rational dynamical system whose formula, degree and the dimension of the space it acts upon depend on the specific algorithm. Nevertheless, we demonstrate that the qualitative behavior of the dynamics exhibit universal features that can be exploited to draw conclusions about the zero sets.

math.DS↗
arXiv2023

On boundedness of zeros of the independence polynomial of tori

We study boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. We prove that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori. Here balanced means that the size of the torus is at most exponential in the shortest side length, while highly unbalanced means that the longest side length of the torus is super exponential in the product over the other side lengths cubed. We discuss implications of our results to the existence of efficient algorithms for approximating the independence polynomial on tori. This project was partially inspired by the relationship between zeros of partition functions and holomorphic dynamics, a relationship that in the last two decades played a prominent role in the field. Besides presenting new results, we survey this relationship and its recent consequences.

math.CO↗
arXiv2021

The limit of the zero locus of the independence polynomial for bounded degree graphs

The goal of this paper is to accurately describe the maximal zero-free region of the independence polynomial for graphs of bounded degree, for large degree bounds. In previous work with de Boer, Guerini and Regts it was demonstrated that this zero-free region coincides with the normality region of the related occupation ratios. These ratios form a discrete semi-group that is in a certain sense generated by finitely many rational maps. We will show that as the degree bound converges to infinity, the properly rescaled normality regions converge to a limit domain, which can be described as the maximal boundedness component of a semi-group generated by infinitely many exponential maps. We prove that away from the real axis, this boundedness component avoids a neighborhood of the boundary of the limit cardioid, answering a recent question by Andreas Galanis. We also give an exact formula for the boundary of the boundedness component near the positive real boundary point.

math.CO↗
arXiv2021

Zeros, chaotic ratios and the computational complexity of approximating the independence polynomial

The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design of efficient algorithms to approximately compute evaluations of the polynomial. In this paper we directly relate the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. We do this by moreover relating the location of zeros to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios.

math.CO↗
arXiv2021

Dynamics of transcendental H\'enon maps III: Infinite entropy

Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental H\'enon maps offers the potential of combining ideas from transcendental dynamics in one variable, and the dynamics of polynomial H\'enon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.

math.DS↗
arXiv2019

Critical intermittency in rational maps

Intermittent dynamics is characterized by long periods of different types of dynamical characteristics, for instance almost periodic dynamics alternated by chaotic dynamics. Critical intermittency is intermittent dynamics that can occur in iterated function systems, and involves a superattracting periodic orbit. This paper will provide and study examples of iterated function systems by two rational maps on the Riemann sphere that give rise to critical intermittency. The main ingredient for this is a superattracting fixed point for one map that is mapped onto a common repelling fixed point by the other map. We include a study of topological properties such as topological transitivity.

math.DS↗
arXiv2019

Automorphisms of $\mathbb C^2$ with parabolic cylinders

A {\sl parabolic cylinder} is an invariant, non-recurrent Fatou component $\Omega$ of an automorphism $F$ of $\mathbb C^2$ satisfying: (1) The closure of the $\omega$-limit set of $F$ on $\Omega$ contains an isolated fixed point, (2) there exists a univalent map $\Phi$ from $\Omega$ into $\mathbb C^2$ conjugating $F$ to the translation $(z,w) \mapsto (z+1, w)$, and (3) every limit map of $\{F^{\circ n}\}$ on $\Omega$ has one-dimensional image. In this paper we prove the existence of parabolic cylinders for an explicit class of maps, and show that examples in this class can be constructed as compositions of shears and overshears.

math.DS↗
arXiv2019

Lee-Yang Zeros of the antiferromagnetic Ising Model

We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising Model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.

math.DS↗
arXiv2019

Dynamics of transcendental H\'enon maps-II

Transcendental H\'enon maps are the natural extensions of the well investigated complex polynomial H\'enon maps to the much larger class of holomorphic automorphisms. We prove here that transcendental H\'enon maps always have non-trivial dynamical behavior, namely that they always admit both periodic and escaping orbits, and that their Julia sets are non-empty and perfect.

math.DS↗
arXiv2019

A transcendental H\'enon map with an oscillating wandering Short $\mathbb{C}^2$

Short $\mathbb{C}^2$'s were constructed in [F] as attracting basins of a sequence of holomorphic automorphisms whose rate of attraction increases superexponentially. The goal of this paper is to show that such domains also arise naturally as autonomous attracting basins: we construct a transcendental H\'enon map with an oscillating wandering Fatou component that is a Short $\mathbb{C}^2$. The superexponential rate of attraction is not obtained at single iterations, but along consecutive oscillations.

math.CV↗
arXiv2019

Coupled fast and slow feedbacks lead to continual evolution: A general modeling approach

The Red Queen Hypothesis, which suggests that continual evolution can result from solely biotic interactions, has been studied in macroevolutionary and microevolutionary contexts. While the latter has been effective in describing examples in which evolution does not cease, describing which properties lead to continual evolution or to stasis remains a major challenge. In many contexts it is unclear which assumptions are necessary for continual evolution, and whether described behavior is robust under perturbations. Our aim here is to prove continual evolution under minimal conditions and in a general framework, thus automatically obtaining robustness. We show that the combination of a fast positive and a slow negative feedback causes continual evolution with a single evolving trait, provided the ecological timescale is sufficiently separated from the timescales of mutations and negative feedback. Our approach and results form a next step towards a deeper understanding of the evolutionary dynamics resulting from biotic interactions.

q-bio.PE↗
arXiv2018

Location of zeros for the partition function of the Ising model on bounded degree graphs

The seminal Lee-Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in $\mathbb C$. In fact the union of the zeros of all graphs is dense on the unit circle. In this paper we study the location of the zeros for the class of graphs of bounded maximum degree $d\geq 3$, both in the ferromagnetic and the anti-ferromagnetic case. We determine the location exactly as a function of the inverse temperature and the degree $d$. An important step in our approach is to translate to the setting of complex dynamics and analyze a dynamical system that is naturally associated to the partition function.

math.CO↗
arXiv2018

A polynomial automorphism with a wandering Fatou component

We construct polynomial automorphisms with wandering Fatou components. The four-dimensional automorphisms $H$ lie in a one-parameter family, depending on the parameter $\delta \in \mathbb C \setminus \{0\}$, and as $\delta \rightarrow 0$ the automorphisms degenerate to the two-dimensional polynomial map $P$ constructed in Astorg et al (Ann. of Math., 2016). Our main result states that if $P$ has a wandering domain, then $H$ does too for $\delta$ sufficiently small.

math.DS↗
arXiv2018

Random local complex dynamics

The study of the dynamics of an holomorphic map near a fixed point is a central topic in complex dynamical systems. In this paper we will consider the corresponding random setting: given a probability measure $\nu$ with compact support on the space of germs of holomorphic maps fixing the origin, we study the compositions $f_n\circ\cdots\circ f_1$, where each $f_i$ is chosen independently with probability $\nu$. As in the deterministic case, the stability of the family of the random iterates is mostly determined by the linear part of the germs in the support of the measure. A particularly interesting case occurs when all Lyapunov indices vanish, in which case stability implies simultaneous linearizability of all germs in $supp(\nu)$.

math.CV↗