In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra (other than tetrahedra) having non-obtuse dihedral angles [4,5]. 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 3dimensional 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". Résumé E. M. Andreev a publié en 1970 une classification des polyèdres hyperboliques compacts de dimension trois (autrement que les tétraèdres) dont les angles dièdres sont non-obtus [4, 5]. Etant donné une description combinatoire d'un polyèdre C, le Théorème d'Andreev dit que les angles dièdres possibles sont exactement décrits par cinq classes d'inégalités linéaires. Le Théorème d'Andreev démontre également que le polyèdre résultant est alors unique à isométrie hyperbolique près. D'une part, le Théorème de Andreev est évidemment un énoncé intéressant de la géométrie de l'espace hyperbolique en dimension 3; d'autre part c'est un outil essentiel dans la preuve du Théorème d'Hyperbolization de Thurston pour les variétés Haken de dimension 3. Il est d'ailleurs remarquable à quel point la démonstration d'Andreev rappelle (en plus simple) la démonstration de Thurston. La démonstration d'Andreev contient une erreur importante. Nous corrigeons ici cette erreur et nous fournissons aussi une nouvelle preuve lisible des autres parties de la preuve, car le papier d'Andreev a la réputation d'être "illisible".

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) =z, where r(z) is a rational function. We study the number of solutions to p(z) =z and r(z) =z, where p(z) and r(z) are polynomials and rational functions, respectively. Upper and lower bounds were previously obtained by Khavinson-Światek, 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) =z describes an n-point gravitational lens, we determine the possible numbers of generic images.

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 R in two variables (the renormalization transformation). We prove that R is partially hyperbolic on an invariant cylinder C. The Lee-Yang zeros are organized in a transverse measure for the central-stable foliation of R| C. Their distribution is absolutely continuous. Its density is C ∞ (and non-vanishing) below the critical temperature. Above the critical temperature, it is C ∞ 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.

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, 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.

Little is known about the global topology of the Fatou set U (f) for holomorphic endomorphisms f : CP k → CP k , when k > 1. Classical theory describes U (f) as the complement in CP k of the support of a dynamically defined closed positive (1, 1) current. Given any closed positive (1, 1) current S on CP k , we give a definition of linking number between closed loops in CP k \ supp S and the current S. It has the property that if lk(γ, S) = 0, then γ represents a non-trivial homology element in H1(CP k \ supp S). As an application, we use these linking numbers to establish that many classes of endomorphisms of 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 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 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.