A Topological Bound on the Cantor–Bendixson Rank of Meromorphic Differentials

We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor–Bendixson rank and give a tight bound. Finally, we present a family of surfaces realizing all possible Cantor–Bendixson ranks. The techniques in the proof of this result exclusively concern Abelian differentials on Riemann surfaces, also known as translation surfaces. The concept of a “slit translation surface” is introduced as the primary tool for studying meromorphic quadratic differentials with higher order poles.

Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy–Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmüller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely linear involutions. These maps are related to general measured foliations on surfaces (whether orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with ℤ/2ℤ linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy–Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows us, in particular, to classify the connected components of all exceptional strata.

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.

We consider various metric and analytic notions of finiteness on translation surfaces. The Veech group of a surface is discrete if the surface has finite area or is totally bounded.

Stress analysis of a finite plate with a rectangular hole subjected to uniaxial tension using modified stress functions

We study the dynamics of the geodesic flow for a class of non-complete Riemannian metrics on a negatively curved surface M . These finite area surfaces are composed of finitely many “singular” surfaces of revolutions of the form y = x, r > 1 for 0 ≤ x ≤ 1 (thin pieces), together with connecting surfaces of bounded negative curvature (thick pieces). The curvature at every point is negative and bounded away from zero, but is unbounded in the “cusps.” The geodesic flow is non-complete because geodesics corresponding to singular unit tangent vectors pointing into the origin hit the cusps in finite time and then cease to be defined. Such singular unit tangent vectors are dense in the unit tangent bundle. Our main result is ergodicity of the geodesic flow (Theorem 5.1). It immediately follows that these metrics have dense geodesics in the unit tangent bundle SM . We also prove that the closed geodesics are dense in the unit tangent bundle (Theorem 6.1) The motivation for considering these metrics arises from studying the geodesic flow for the Weil-Petersson (WP) metric on two dimensional moduli spaces of Riemann surfaces., e.g., the moduli space for the once punctured torus. In several fundamental ways the geodesic flow we consider provides a good model for the WP geodesic flow. For example, Wolpert showed that the WP metric is also non-complete, has finite area, has negative curvature bounded away from zero, and is unbounded in the ”cusps.” Also, various authors have shown that the cusps can be approximated by singular surface of revolutions. At present, there is only a single technical obstruction (additional estimates on the derivatives of the GF in the cusp) that prevent us from applying this general method to prove ergodicity of the WP geodesic flow. See Section 7. The geometric properties of both types of flows imply that the geodesic flow is a uniformly hyperbolic dynamical system with singularities. The non-completeness causes pathologies in the stable and unstable manifolds, as for many billiard flows. For example, stable and unstable manifolds at a point, if they exist, may intersect a cusp point, and thus have only finite length. One needs extensions of “Pesin theory” to systems with singularities to study these geodesic flows. Another challenging aspect is that these surfaces may be simply connected, making the use of any of the traditional arguments involving boundaries of the covering spaces inapplicable. Studying the dynamics requires the development of new approches. Our strategy to prove ergodicity of the geodesic flow is to first establish non-uniform hyperbolicity, i.e., the Lyapunov exponents are non-zero at almost every point. We then prove that there

We define ranks and degrees for families of theories, similar to Morley rank and degree, as well as Cantor–Bendixson rank and degree, and the notion of totally transcendental family of theories. Bounds for $$e$$ -spectra with respect to ranks and degrees are found. It is shown that the ranks and the degrees are preserved under $$E$$ -closures and values for the ranks and the degrees are characterized. Criteria for totally transcendental families in terms of cardinality of $$E$$ -closure and of the $$e$$ -spectrum value, for a countable language, are proved.

Review of (Z.2) on the compact case Trace formula for Tr *sRr j (*s cuspidal) Trace formula for Tr*sRr j (*s Einstein) Spectral estimates (Weyl laws) Asymptotic expansions (Prime geodesic theorems) Equidistribution of closed goedesics.

We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary. We also prove finiteness results concerning cylinders, a partial converse to the Cylinder Deformation Theorem, and a result generalizing part of the Veech dichotomy.

We show that for every weakly compact subset K of C[0, 1] with finite Cantor–Bendixson rank, there is a reflexive Banach lattice E and an operator \(T:E\rightarrow C[0,1]\) such that \(K\subseteq T(B_E)\). On the other hand, we exhibit an example of a weakly compact set of C[0, 1] homeomorphic to \(\omega ^\omega +1\) for which such T and E cannot exist. This answers a question of M. Talagrand in the 80’s.

The torsion-free part of the Ziegler spectrum of the Klein four group

The space of marked commutative rings on n given generators is a compact metrizable space. We compute the Cantor–Bendixson rank of any member of this space. For instance, the Cantor–Bendixson rank of the free commutative ring on n generators is ωn, where ω is the smallest infinite ordinal. More generally, we work in the space of finitely generated modules over a given commutative ring.

We demonstrate how several programming language concepts and methods can be used economically to obtain an improved solution to a difficult algorithmic problem. The problem is to compile a subset RCS of Relational Calculus defined by Willard (1978) in a novel way so that efficient run-time query performance is guaranteed. Willard gives an algorithm to compile each query q belonging to RCS so that it executes in O(n logd n + o) steps and O(n) space, where n and o are respectively the input and output set sizes, and d is a parameter associated with the syntax of query q. Willard’s time bounds are based on the assumption that hashing unit-space data takes unit time.In this paper we use a set-theoretic complexity measure and formal transformational techniques to reconstruct the linear time fragment of RCS in a simplified way. In doing this, we show how complexity can be determined by language abstraction and algebraic reasoning without resorting to low level counting arguments. This approach shortens Willard’s proofs considerably, and facilitates an implementation. Finally, we show that the implementation can be typed in a restricted form of a set-theoretic type system based on Cai et al. (1991) and Keller and Paige (1995), which guarantees that each hash operation used to implement RCS can be simulated in real-time on a pointer machine model of computation. This improves Willard’s linear expected time result to linear worst case time, and demonstrates for the first time how type theory can be used as a tool to obtain an order of magnitude algorithmic speedup.KeywordsInference RuleQuery GraphQuantifier EliminationPrimitive OperationPointer MachineThese 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.

Let $S$ be a Riemann surface of finite area with at least one puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, the pair $(a, f^m(a))$ of geodesics fills $S$ for $m\geq 3$. We also study the cases of $0<m\leq 2$ and show that if $(a,f^2(a))$ does not fill $S$, then there is only one geodesic $b$ such that $b$ is disjoint from both $a$ and $f^2(a)$. In fact, $b=f(a)$ and $\{a,f(a)\}$ forms the boundary of an $x$-punctured cylinder on $S$. As a consequence, we show that if $a$ and $f(a)$ are not disjoint, then $(a,f^m(a))$ fills $S$ for any $m\geq 2$.

On the compactness and closure of surfaces of finite area, continuous or otherwise, and on generalized surfaces