Generalized Staircases: Recurrence and Symmetry

We study the symmetries and geodesics of an infinite translation surface which arises as a limit of translation surfaces built from regular polygons, studied by Veech. We find the affine symmetry group of this infinite translation surface, and we show that this surface admits a deformation into other surfaces with topologically equivalent affine symmetries. The geodesics on these new surfaces are combinatorially the same as the geodesics on the original.

This thesis uses dynamical systems to study two types of objects: number expansions (Chapters 1 and 2) and translation surfaces (Chapter 3). Our first chapter builds a broad, unifying theory for a large class of continued fraction algorithms producing what we call contracted Farey expansions. These algorithms are based on three ideas: (i) contraction of generalised continued fractions, (ii) induced transformations, and (iii) the natural extension of the Farey tent map. Within this theory, we find several well-studied algorithms; a new subfamily of superoptimal continued fractions with arbitrarily good convergence and approximation properties; and a unifying framework to prove several old and new results in Diophantine approximation. Chapter 2 introduces a new, one-parameter family of functions called skewed symmetric golden maps. Using tools from ergodic theory, we study the relative frequencies of digits typically occurring in number expansions produced by these maps. The central tool for our analysis is a mysterious phenomenon of our functions called matching, which has been recently observed and exploited to understand several other families of functions generating number expansions. Our final chapter deals with translation surfaces, i.e., surfaces obtained by gluing pairs of parallel, equal-length, and oppositely oriented edges of planar polygons. The Veech group of a translation surface is the group of Jacobians of its orientation-preserving affine automorphisms. We develop a novel algorithm to construct translation surfaces with prescribed lattice Veech groups in given strata. Our ideas are also used to obtain obstructions for the realisability of certain Veech groups in certain strata. In particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is the modular group.

The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all translation surfaces with a given lattice Veech group in any given stratum. In developing this algorithm, we give a new proof of a finiteness result of Smillie and Weiss, namely that there are only finitely many unit-area translation surfaces in any stratum with the same lattice Veech group. Our methods can be applied to obtain obstructions of lattices being realized as Veech groups in certain strata; in particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is all of textrm{SL}_2mathbb {Z}.

We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions for which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than 1/2. We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multi-twists in a pair of cylinder decompositions in non-parallel directions.

In this paper we study the structure of non-trivial recurrence leaves. The obtained results have been shown to be very useful in the smoothing of the continuous singular foliations on arbitrary genus two-manifolds [Lo1]. From the famous example of Denjoy [De], it is clear that it is very important to understand the dynamical structure of non-trivial recurrence. For compact two-manifolds, the study of non-trivial recurrence together with the smoothing problems for orientable singular foliations was carried out by C. Gutierrez [Gu1]. On the other hand, recent work shows that infinite genus surfaces can support orientable foliations whose recurrent leaves have dynamics which can not appear on compact surfaces (see [Gu-He-Lo]). Besides, the existence of minimal foliations on those surfaces was already proven by J. Beniere (see [Be]). Our main theorem generalize, to orientable singular foliations on two-manifolds of infinite genus, C. Gutierrez’s structure theorem [Gu1]. To state the theorem, we need some definitions. Let T : R/Z→ R/Z be a map whose domain of definition (Dom(T )) and image (Im(T )) are open and dense subsets of R/Z. We say that T is a generalized interval exchange transformation(or shortly a GIET) if T takes homeomorphically each connected component of its domain of definition onto a connected component of its image. A GIET is said to be affine (resp. isometric) if it restricted to every connected component of its domain of definition, is affine (resp. isometric). If T is an isometric GIET such that R/Z Dom(T ) is at most finite, then we shall say that T is a standard interval exchange transformation (standard IET).

In view of classical results of Masur and Veech almost every element in the moduli space of compact translation surfaces is recurrent. In this paper we focus on the problem of recurrence for elements of smooth curves in the moduli space. We give an effective criterion for the recurrence of almost every element of a smooth curve. The criterion relies on results developed by Minsky-Weiss in \cite{Mi-We}. Next we apply the criterion to the billiard flow on planar tables confined by arcs of confocal conics. The phase space of such billiard flow splits into invariant subsets determined by caustics. We prove that for almost every caustic the billiard flow restricted to the corresponding invariant set is uniquely ergodic. This answers affirmatively to a question raised by Zorich.

Forced oscillations of a massive point on a compact surface with a boundary

We compute the first variation of the functional that assigns each\nunit vector field the volume of its image in the unit tangent bundle.\nIt is shown that critical points are exactly those vector fields that\ndetermine a minimal immersion. We also find a necessary and\nsufficient condition that a vector field, defined in an open manifold,\nmust fulfill to be minimal, and obtain a simpler equivalent condition\nwhen the vector field is Killing. The condition is fulfilled, in particular,\nby the characteristic vector field of a Sasakian manifold and by Hopf vector\nfields on spheres.

We analyze a least-squares strong-form kernel collocation formulation for solving second-order elliptic PDEs on smooth, connected, and compact surfaces with bounded geometry. The methods do not require any partial derivatives of surface normal vectors or metric. Based on some standard smoothness assumptions for high-order convergence, we provide the sufficient denseness conditions on the collocation points to ensure the methods are convergent. In addition to some convergence verifications, we also simulate some reaction-diffusion equations to exhibit the pattern formations.

Locally concentrated solutions in the Born–Infeld theory are presented. In particular, existence and uniqueness theorems are established for multicentred magnetic string solutions induced from a Higgs field over a closed Riemann surface or a Euclidean plane. On any given compact surface, the Born–Infeld parameter may be adjusted under a necessary and sufficient condition to allow the existence of an arbitrarily large number of strings.

Commutativity and Wecken properties for fixed points on surfaces and 3-manifolds

In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [Calabi], while the solution to the local problem was found by [Wallach]. The case of nonpositive Gaussian curvature is more subtle and has remained open. We prove that there are no minimal surfaces in Sn of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into S. 0. Introduction. In this paper, we classify the connected minimal surfaces of constant Gaussian curvature in the unit n-sphere Sn C Enll for all n. It is a classical fact (and follows easily from the structure equations) that the only examples up to rigid motion in S3 are the open subsets of the geodesic spheres (with K 1) and the Clifford torus (with K 0). In an early paper Boruvka constructed a linearly full immersion S2 C S2m for each m > 0, where the induced metric had K 2/m(m + 1). Later [Calabi] showed that, up to rigid motion, these Boruvka spheres were the only compact minimal surfaces with K Ko > 0 in S' for any n. [Wallach] proved that any connected piece of minimal surface with K a positive constant in Sn was a subset of a Boruvka sphere. [Kenmotsu, 1976] found all of the flat minimal surfaces (K 0) in S' and quite recently, [Kenmotsu, 1983] showed that K Ko < 0 is impossible for minimal surfaces in S4. The techniques used in the above proofs range from harmonic analysis to rather involved calculations with the moving frame. On the other hand, in this paper, we take a somewhat different point of view. If (M2, ds2) is a surface of constant Gaussian curvature K, then the minimal surfaces in S n C E n ? 1 of constant Gaussian curvature K are given locally by smooth maps f: M En' + lwhich satisfy three conditions: first, Kf, f) 1; second, Af =2f (A is the Laplace-Beltrami operator of ds2); and third, that f should be an isometry, Kdf, df) = ds2. We study the more general class of maps f: M En + 1 which only satisfy the first two conditions. Note that this set of equations is already overdetermined (by one equation). Received by the editors August 14, 1984. 1980 Mathematics Subject Classification. Primary 53A10; Secondary 53B25 53C40.

A necessary and sufficient condition is given for a nontrivial homology class of a simply connected 4-manifold to be represented by a simple, topologically locally flat embedding of a compact Riemann surface. This result extends the embedding theorem of the 2-sphere, proved by the authors in K-Theory 7 (1993), to the case of a surface of any genus. As a consequence, the computation of the least genus of a locally flat surface is obtained for any homology class of even divisibility. The computation of the least genus is also settled for certain (and perhaps all) classes of odd divisibility. In the case of a minimum genus surface it is shown that the constructed embedding is often nonsmoothable. For example, all but nine homology classes in the complex projective plane contain such a nonsmoothable surface. A similar result holds for any class of nonnegative self intersection in a K3 surface.

We investigate the gradient flow associated to the prescribed scalar curvature problem on compact Riemannian surfaces. We prove the global existence and the convergence at infinity of this flow under sufficient conditions on the prescribed function, which we suppose just continuous. In particular, this gives a uniform approach to solve the prescribed scalar curvature problem for general compact surfaces.

Many real-time embedded systems involve a collection of independently executing event-driven code blocks, having hard real-time constraints. Portions of such codes when triggered by external events require to be executed within a given deadline from the triggering time. The feasibility analysis problem for such a real-time system asks whether it is possible to schedule all such blocks of code so that all the associated deadlines are met even in the worst case triggering sequence. Each such conditional real-time code block can be naturally represented by a directed acyclic graph whose vertices correspond to portions of code having a straight-line flow of control and are associated with execution requirements and deadlines relative to their triggering times, and the edges represent conditional branches. Till now, no complexity results were known for the feasibility analysis problem in this model, and all existing algorithms in the real-time systems literature have an exponential complexity. In this paper we show that this problem is NP-hard under both dynamic and static priorities in the preemptive uniprocessor case, even for a set of only two task graphs. For dynamic-priority feasibility analysis we give a pseudo-polynomial time exact algorithm and a fully polynomial-time approximation scheme for approximate feasibility testing. For the special case where all the execution requirements of the vertices are identical, we present a polynomial time exact algorithm. For static-priority feasibility analysis, we introduce a new sufficient condition and give a pseudo-polynomial time algorithm for checking it. This algorithm gives tighter results for feasibility analysis compared to those known so far.KeywordsCode BlockTask GraphMinimum Time IntervalFeasibility AnalysisConditional BranchThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.