Eugene Gutkin

Affine mappings of translation surfaces: geometry and arithmetic

1. Introduction. Translation surfaces naturally arise in the study of billiards in rational polygons (see [ZKa]). To any such polygon P , there corresponds a unique translation surface, S = S(P), such that the billiard flow in P is equivalent to the geodesic flow on S (see, e.g., [Gu2], [Gu3]). There is also a classical relation between translation surfaces and quadratic differentials on a Riemann surface S. Namely, each quadratic differential induces a translation structure on a finite puncturing of S or on a canonical double covering of S. Quadratic differentials have a natural interpretation as cotangent vectors to Teich-müller space, and this connection has proven useful in the study of billiards (see, e.g., [Ma2], [V1]). With a translation surface, S, one associates various algebraic and geometric objects: the induced affine structure of S and the group of affine diffeomorphisms, Aff(S); the holonomy homomorphism, hol : π 1 (S) → R 2 and the holonomy group Hol(S) = hol(π 1 (S)); the flat structure on S and the natural cell decompositions of its metric completion S. In the present paper, we study the relations between these objects, as well as relations among different translation surfaces. Our main focus is the group Aff(S) and the associated group of differentials, (S) ⊂ SL(2, R). The study of these groups began as part of W. Thurstons classification of surface diffeomorphisms in [Th2]. This study continued with the work of W. Veech in [V1] and [V2]. Veech produced explicit examples of translation surfaces S for which (S) is a nonarithmetic lattice. He showed that if (S) is a lattice, then the geodesic flow on S exhibits remarkable dynamical properties. For these reasons, we call (S) the Veech group of S, and if this group is a lattice, then we call S a Veech surface. We now describe the structure of the paper and our main results. In §2, we establish the setting. In particular, we recall the notion of a G-manifold and associated objects: the developing map, the holonomy homomorphism, and the holonomy group. We introduce the notion of the differential of a G-map with respect to a normal subgroup H ⊂ G. We also introduce the spinal triangulation, one of several cell decompositions canonically associated to a flat surface with cone points. In §3, we study G-coverings of G-manifolds. Given such a covering, p : X → Y , we characterize the group …

Billiards in polygons: Survey of recent results

We review the dynamics of polygonal billiards

Caustics for inner and outer billiards

With a plane closed convex curve,T, we associate two area preserving twist maps: the (classical) inner billiard inT and the outer billiard in the exterior ofT. The invariant circles of these twist maps correspond to certain plane curves: the inner and the outer caustics ofT. We investigate how the shape ofT determines the possible location of caustics, establish the existence of open regions which are free of caustics, and estimate fro below the size of these regions in terms of the geometry ofT.

Billiards on almost integrable polyhedral surfaces

The phase space of the geodesic flow on an almost integrable polyhedral surface is foliated into a one-parameter family of invariant surfaces. The flow on a typical invariant surface is minimal. We associate with an almost integrable polyhedral surface its holonomy group which is a subgroup of the group of motions of the Euclidean plane. We show that if the holonomy group is discrete then the flow on an invariant surface is ergodic if and only if it is minimal.

Dual polygonal billiards and necklace dynamics

We study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics. Our main result is that for a certain class of tables, called the quasi-rational polygons, the dual billiard orbits are bounded. This implies that for the subset of rational tables (i.e. polygons with rational vertices) the dual billiard orbits are periodic.

Topological entropy of generalized polygon exchanges

We obtain geometric upper bounds on the topological entropy of generalized polygon exchange transformations. As an application of our results, we show that billiards in polygons and rational polytopes have zero topological entropy.

Topological entropy of polygon exchange transformations and polygonal billiards

We study the topological entropy of a class of transformations with mild singularities: the generalized polygon exchanges. This class contains, in particular, polygonal billiards. Our main result is a geometric estimate, from above, on the topological entropy of generalized polygon exchanges. One of the applications of our estimate is that the topological entropy of polygonal billiards is zero. This implies the subexponential growth of various geometric quantities associated with a polygon. Other applications are to the piecewise isometries in two dimensions, and to billiards in rational polyhedra.

DIFFERENTIAL TOPOLOGY OF NUMERICAL RANGE

Abstract The numerical range of an n × n matrix, also known as its field of values , is reformulated as the image of a smooth quadratic mapping from the n − 1 dimensional complex projective space to the complex plane. This paper investigates the numerical range from the perspective of differential topology (Morse theory). More specifically, the boundary of the range is interpreted as a rank 1 critical value curve and its sharp points are interpreted as rank 0 critical values. More importantly, the map is shown to have additional critical value curves in the interior of the numerical range. These additional curves are shown to have such singularity phenomena as cusps and swallow tails, to be the caustic envelopes of families of lines, and to exhibit the so-called “normal bifurcation” when an eigenvalue becomes unitarily decoupled.

Quantum nonlinear Schrödinger equation: Two solutions

Abstract The quantum nonlinear Schrodinger equation (QNLS) has attracted much attention recently as a simplest exactly soluble nonlinear model of the quantum field theory in 1 + 1 space-time. There are two approaches in the literature to solution of QNLS. One is a quantization prescription for the solution of classical nonlinear Schrodinger equation by the inverse scattering method. It is called the quantum inverse method (QIM). The other is an elaboration of the Bethe Ansatz technique. It is called the method of intertwining operators (MIO). The two approaches produce formally different expansions for the QNLS field and for certain Fock space operators associated with it. In this work we give a comparative exposition of both methods. We then show that the QIM expansions and the MIO expansions define the same operators on Fock space.

Integrable Hamiltonians with exponential potential

We construct a large class of integrable Hamiltonian systems with n degrees of freedom. This class naturally extends the nonperiodic Hamiltonians of Toda lattice type.