Poul G. Hjorth

Finite metric spaces of strictly negative type

Abstract We prove that, if a finite metric space is of strictly negative type, then its transfinite diameter is uniquely realized by the infinite extender (load vector). Finite metric spaces that have this property include all spaces on two, three, or four points, all trees, and all finite subspaces of Euclidean spaces. We prove that, if the distance matrix is both hypermetric and regular, then it is of strictly negative type. We show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points. In connection with an open problem raised by Kelly, we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular, and hence of strictly negative type.

Hyperbolic spaces are of strictly negative type

We study finite metric spaces with elements picked from, and distances consistent with, ambient Riemannian manifolds. The concepts of negative type and strictly negative type are reviewed, and the conjecture that hyperbolic spaces are of strictly negative type is settled, in the affirmative. The technique of the proof is subsequently applied to show that every compact manifold of negative type must have trivial fundamental group, and to obtain a necessary criterion for product manifolds to be of negative type.

Real and complex dynamical systems

Preface. Dynamical Zeta Functions V. Baladi. The Global Dynamics of Impact Oscillators C. Budd. Grazing in Impact Oscillators C. Budd. Topological Entropy of Unimodal Maps A. Douady. Henon Mappings in the Complex Domain J. H. Hubbard, R. W. Oberste-Vorth. Symbolic Dynamics, Group Automorphisms and Markov Partitions B. Kitchens. A Monotonicity Conjecture for Real Cubic Maps S. P. Dawson, R. Galeeva, J. Milnor, C. Tresser. Dynamics of Ordinary Differential Equations C. Sparrow. Real Bounds in Complex Dynamics S. van Strien. Homoclinic Bifurcations and Strange Attractors M. Viana. Introduction to Hyperbolic Dynamics J.-C. Yoccoz. Ergodic Theory of Differentiable Dynamical Systems Lai-Sang Young. Index.

Dynamics on the Riemann Sphere

Branner-Hubbard motion is a systematic way of deforming an attracting holomorphic dynamical system f into a family (fs)s∈L, via a holomorphic motion which is also a group action. We establish the analytic dependence of fs on s (a result first stated by Lyubich) and the injectivity of fs on f . We prove that the stabilizer of f (in terms of s) is either the full group L (rigidity), or a discrete subgroup (injectivity). The first case means that fs is Mobius conjugate to f for all s∈L, and it happens for instance at the center of a hyperbolic component. In the second case the map s → fs is locally injective. We show that BH-motion induces a periodic holomorphic motion on the parameter space of cubic polynomials, and that the corresponding quotient motion has a natural extension to its isolated singularity. We give another application in the setting of Lavaurs enriched dynamical systems within a parabolic basin.We consider the Arnold family of analytic diffeomorphisms of the circle x 7! x + t + a 2� sin(2�x) mod (1), where a,t 2 (0,1) and its complexification f�,a(z) = �ze a 2 (z 1 z ) , with � = e 2�it a holomorphic self map of C � . The parameter space contains the well known Arnold tongues Tfor� 2 (0,1) being the rotation number. We are interested in the parameters that belong to the irrational tongues and in particular in those for which the map has a Herman ring. Our goal in this paper is twofold. First we are interested in studying how the modulus of this Herman ring varies in terms of the parametera, when a tends to 0 along the curve T�. We survey the different results that describe this variation including the complexification of part of the Arnold tongues (called Arnold disks) which leads to the best estimate. To work with this complex parameter values we use the concept of the twist coordinate, a measure of how far from symmetric the Herman rings are. Our second goal is to investigate the slice of parameter space that contains all maps in the family with twist coordinate equal to one half, proving for example that this is a plane in C 2 . We show a computer picture of this slice of parameter space and we also present some numerical algorithms that allow us to compute new drawings of non-symmetric Herman rings of various moduli.An exposition of the 1918 paper of Lattès, together with its historical antecedents, and its modern formulations and applications. 1. The Lattès paper. 2. Finite Quotients of Affine Maps 3. A Cyclic Group Action on C/Λ . 4. Flat Orbifold Metrics 5. Classification 6. Lattès Maps before Lattès 7. More Recent Developments 8. Examples References §1. The Lattès paper. In 1918, some months before his death of typhoid fever, Samuel Lattès published a brief paper describing an extremely interesting class of rational maps. Similar examples had been described by Schröder almost fifty years earlier (see §6), but Lattès’ name has become firmly attached to these maps, which play a basic role as exceptional examples in the holomorphic dynamics literature. His starting point was the “Poincaré function” θ : C → Ĉ associated with a repelling fixed point z0 = f(z0 ) of a rational function f : Ĉ → Ĉ . This can be described as the inverse of the Kœnigs linearization around z0 , extended to a globally defined meromorphic function.1 Assuming for convenience that z0 ̸= ∞ , it is characterized by the identity f(θ(t)) = θ(μ t) for all complex numbers t , with θ(0) = z0 , normalized by the condition that θ′(0) = 1 . Here μ = f ′(z0 ) is the multiplier at z0 , with |μ| > 1 . This Poincaré function can be computed explicitly by the formula θ(t) = lim n→∞ f ◦n ( z0 + t/μ n ) . Its image θ(C) ⊂ Ĉ is equal to the Riemann sphere Ĉ with at most two points removed. In practice, we will always assume that f has degree at least two. The complement Ĉ ! θ(C) is then precisely equal to the exceptional set Ef , consisting of all points with finite grand orbit under f . In general this Poincaré function θ has very complicated behavior. In particular, the Poincaré functions associated with different fixed points or periodic points are usually quite incompatible. However, Lattès pointed out that in special cases θ will be periodic or doubly periodic, and will give rise to a simultaneous linearization for all of the periodic points of f . (For a more precise statement, see the proof of 3.9 below.) 1 Compare [La], [P], [K]. For general background material, see for example [M3] or [BM].

Equation-free detection and continuation of a Hopf bifurcation point in a particle model of pedestrian flow

Using an equation-free analysis approach we identify a Hopf bifurcation point and perform a two-parameter continuation of the Hopf point for the macroscopic dynamical behavior of an interacting particle model. Due to the nature of systems with a moderate number of particles and noise, the quality of the available numerical information requires the use of very robust numerical algorithms for each of the building blocks of the equation-free methodology. As an example, we consider a particle model of a crowd of pedestrians where particles interact through pairwise “social forces.” The pedestrians move along a corridor where they are constrained by the walls of the corridor, and two crowds are aiming, from opposite directions, to pass through a narrowing doorway perpendicular to the corridor. We focus our investigation on the collective behavior of the model. As the width of the doorway is increased, we observe an onset of oscillations of the net pedestrian flux through the doorway, described by a Hopf bifurc...

Exponentially Long Equilibrium Times in a One-Dimensional Collisional Model of Classical Gas

Around 1900, J. H. Jeans suggested that the “abnormal” specific heats observed in diatomic gases, specifically the lack of contribution to the heat capacity from the internal vibrational degrees of freedom, in apparent violation of the equipartition theorem, might be caused by the large separation between the time scale for the vibration and the time scale associated with a typical binary collision in the gas. We consider here a simple 1D model and show how, when these time scales are well separated, the collisional dynamics is constrained by a many- particle adiabatic invariant. The effect is that the collisional energy exchanges betgween the translational and the vibrational degrees of freedom are slowed down by an exponential factor (as Jeans conjectured). A metastable situation thus occurs, in which the fast vibrational degrees of freedom effectively do not contribute to the specific heat. Hence, the observed “freezing out” of the vibrational degrees of freedom could in principle be explained in terms of classical mechanics. We discuss the phenomenon analytically, on the basis of an approximation introduced by Landau and Teller (1936) for a related phenomenon, and estimate the time scale for the evolution to statistical equilibrium. The theoretical analysis is supported by numerical examples.

Integrated Inflammatory Stress (ITIS) Model

During the last decade, there has been an increasing interest in the coupling between the acute inflammatory response and the Hypothalamic–Pituitary–Adrenal (HPA) axis. The inflammatory response is activated acutely by pathogen- or damage-related molecular patterns, whereas the HPA axis maintains a long-term level of the stress hormone cortisol which is also anti-inflammatory. A new integrated model of the interaction between these two subsystems of the inflammatory system is proposed and coined the integrated inflammatory stress (ITIS) model. The coupling mechanisms describing the interactions between the subsystems in the ITIS model are formulated based on biological reasoning and its ability to describe clinical data. The ITIS model is calibrated and validated by simulating various scenarios related to endotoxin (LPS) exposure. The model is capable of reproducing human data of tumor necrosis factor alpha, adrenocorticotropic hormone (ACTH) and cortisol and suggests that repeated LPS injections lead to a deficient response. The ITIS model predicts that the most extensive response to an LPS injection in ACTH and cortisol concentrations is observed in the early hours of the day. A constant activation results in elevated levels of the variables in the model while a prolonged change of the oscillations in ACTH and cortisol concentrations is the most pronounced result of different LPS doses predicted by the model.

Dynamics on the Riemann Sphere: A Bodil Branner Festschrift

Branner-Hubbard motion is a systematic way of deforming an attracting holomorphic dynamical system f into a family (fs)s∈L, via a holomorphic motion which is also a group action. We establish the analytic dependence of fs on s (a result first stated by Lyubich) and the injectivity of fs on f . We prove that the stabilizer of f (in terms of s) is either the full group L (rigidity), or a discrete subgroup (injectivity). The first case means that fs is Mobius conjugate to f for all s∈L, and it happens for instance at the center of a hyperbolic component. In the second case the map s → fs is locally injective. We show that BH-motion induces a periodic holomorphic motion on the parameter space of cubic polynomials, and that the corresponding quotient motion has a natural extension to its isolated singularity. We give another application in the setting of Lavaurs enriched dynamical systems within a parabolic basin.We consider the Arnold family of analytic diffeomorphisms of the circle x 7! x + t + a 2� sin(2�x) mod (1), where a,t 2 (0,1) and its complexification f�,a(z) = �ze a 2 (z 1 z ) , with � = e 2�it a holomorphic self map of C � . The parameter space contains the well known Arnold tongues Tfor� 2 (0,1) being the rotation number. We are interested in the parameters that belong to the irrational tongues and in particular in those for which the map has a Herman ring. Our goal in this paper is twofold. First we are interested in studying how the modulus of this Herman ring varies in terms of the parametera, when a tends to 0 along the curve T�. We survey the different results that describe this variation including the complexification of part of the Arnold tongues (called Arnold disks) which leads to the best estimate. To work with this complex parameter values we use the concept of the twist coordinate, a measure of how far from symmetric the Herman rings are. Our second goal is to investigate the slice of parameter space that contains all maps in the family with twist coordinate equal to one half, proving for example that this is a plane in C 2 . We show a computer picture of this slice of parameter space and we also present some numerical algorithms that allow us to compute new drawings of non-symmetric Herman rings of various moduli.An exposition of the 1918 paper of Lattès, together with its historical antecedents, and its modern formulations and applications. 1. The Lattès paper. 2. Finite Quotients of Affine Maps 3. A Cyclic Group Action on C/Λ . 4. Flat Orbifold Metrics 5. Classification 6. Lattès Maps before Lattès 7. More Recent Developments 8. Examples References §1. The Lattès paper. In 1918, some months before his death of typhoid fever, Samuel Lattès published a brief paper describing an extremely interesting class of rational maps. Similar examples had been described by Schröder almost fifty years earlier (see §6), but Lattès’ name has become firmly attached to these maps, which play a basic role as exceptional examples in the holomorphic dynamics literature. His starting point was the “Poincaré function” θ : C → Ĉ associated with a repelling fixed point z0 = f(z0 ) of a rational function f : Ĉ → Ĉ . This can be described as the inverse of the Kœnigs linearization around z0 , extended to a globally defined meromorphic function.1 Assuming for convenience that z0 ̸= ∞ , it is characterized by the identity f(θ(t)) = θ(μ t) for all complex numbers t , with θ(0) = z0 , normalized by the condition that θ′(0) = 1 . Here μ = f ′(z0 ) is the multiplier at z0 , with |μ| > 1 . This Poincaré function can be computed explicitly by the formula θ(t) = lim n→∞ f ◦n ( z0 + t/μ n ) . Its image θ(C) ⊂ Ĉ is equal to the Riemann sphere Ĉ with at most two points removed. In practice, we will always assume that f has degree at least two. The complement Ĉ ! θ(C) is then precisely equal to the exceptional set Ef , consisting of all points with finite grand orbit under f . In general this Poincaré function θ has very complicated behavior. In particular, the Poincaré functions associated with different fixed points or periodic points are usually quite incompatible. However, Lattès pointed out that in special cases θ will be periodic or doubly periodic, and will give rise to a simultaneous linearization for all of the periodic points of f . (For a more precise statement, see the proof of 3.9 below.) 1 Compare [La], [P], [K]. For general background material, see for example [M3] or [BM].

De novo generation of molecular structures using optimization to select graphs on a given lattice.

A recurrent problem in organic chemistry is the generation of new molecular structures that conform to some predetermined set of structural constraints that are imposed in an endeavor to build certain required properties into the newly generated structure. An example of this is the pharmacophore model, used in medicinal chemistry to guide de novo design or selection of suitable structures from compound databases. We propose here a method that efficiently links up a selected number of required atom positions while at the same time directing the emergent molecular skeleton to avoid forbidden positions. The linkage process takes place on a lattice whose unit step length and overall geometry is designed to match typical architectures of organic molecules. We use an optimization method to select from the many different graphs possible. The approach is demonstrated in an example where crystal structures of the same (in this case rigid) ligand complexed with different proteins are available.

Rotation of a Rigid Body

The general treatment of the motion of a rigid body is rather involved and we shall in this book consider only certain special cases. The insights to be gained can be applied in virtually all of theoretical physics, and have important technological ramifications. Applications of the theorems derived stretch from studies of the spin of the electron and rotating atomic nuclei, to investigations of the motion of planets and galaxies.