Saddle Connections Research Articles

Backward Rauzy-Veech algorithm and horizontal saddle connections

The goal of this note is to study the action of the backward Rauzy-Veech algorithm on the translation surfaces with horizontal saddle connections. In particular, we prove that the orbit of a translation surface via the aforementioned algorithm is $\infty$-complete if and only if the surface does not posses horizontal saddle connections. Moreover, we show that appearance of all saddle connections as sides of the polygonal representations of the translations surface along the backward Rauzy-Veech induction orbit is equivalent to the minimality of the horizontal translation flow.

Shrinking rates of horizontal gaps for generic translation surfaces

A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\\rightarrow \\infty $$\\end{document} for the difference in angle between two almost horizontal saddle connections.

Horospherical dynamics in invariant subvarieties

We consider the horospherical foliation on any invariant subvariety in the moduli space of translation surfaces. This foliation can be described dynamically as the strong unstable foliation for the geodesic flow on the invariant subvariety, and geometrically, it is induced by the canonical splitting of C-valued cohomology into its real and imaginary parts. We define a natural volume form on the leaves of this foliation, and define horospherical measures as those measures whose conditional measures on leaves are given by the volume form. We show that the natural measures on invariant subvarieties, and in particular, the Masur-Veech measures on strata, are horospherical. We show that these measures are the unique horospherical measures giving zero mass to the set of surfaces with horizontal saddle connections, extending work of Lindenstrauss-Mirzakhani and Hamenstädt for principal strata. We describe all the leaf closures for the horospherical foliation.

Седловые связки

We prove that all vector fields that close to a field with the same saddle connections form a smooth Banach submanifold. Provide a sufficient condition for appearing of saddle connections in a generic family. Also we prove that after a generic perturbation of a monodromial hyperbolic polycycle formed by n saddle connections at least n limit cycles appear.

Counting pairs of saddle connections

We show that for almost every translation surface the number of pairs of saddle connections with bounded magnitude of the cross product has asymptotic growth like cR2 where the constant c depends only on the area and the connected component of the stratum. The proof techniques combine classical results for counting saddle connections with the crucial result that the Siegel–Veech transform is in L2. In order to capture information about pairs of saddle connections, we consider pairs with bounded magnitude of the cross product since the set of such pairs can be approximated by a fibered set which is equivariant under geodesic flow. In the case of lattice surfaces, small bounded magnitude of the cross product is equivalent to counting parallel pairs of saddle connections, which also have a quadratic growth of cR2 where c depends in this case on the given lattice surface.

Impact of dissipation ratio on vanishing viscosity solutions of the Riemann problem for chemical flooding model

The solutions for a Riemann problem arising in chemical flooding models are studied using vanishing viscosity as an admissibility criterion. We show that when the flow function depends non-monotonically on the concentration of chemicals, non-classical undercompressive shocks appear. The monotonic dependence of the shock velocity on the ratio of dissipative coefficients is proven. For that purpose we provide the classification of the nullcline configurations for the traveling wave dynamical systems and analyze the saddle–saddle connections.

Measure bound for translation surfaces with short saddle connections

Measure bound for translation surfaces with short saddle connections

Horizon saddle connections and Morse–Smale dynamics of dilation surfaces

Horizon saddle connections and Morse–Smale dynamics of dilation surfaces

Slope gap distribution of saddle connections on the <inline-formula><tex-math id="M1">$ 2n $</tex-math></inline-formula>-gon

<p style='text-indent:20px;'>We explicitly compute the limiting slope gap distribution for saddle connections on any <inline-formula><tex-math id="M2">\begin{document}$ 2n $\end{document}</tex-math></inline-formula>-gon for <inline-formula><tex-math id="M3">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>. Our calculations show that the slope gap distribution for a translation surface is not always unimodal, answering a question of Athreya. We also give linear lower and upper bounds for number of non-differentiability points as <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> grows. The latter result exhibits the first example of a non-trivial bound on an infinite family of translation surfaces and answers a question by Kumanduri-Sanchez-Wang.</p>

Global dynamics of an amensalism system with Michaelis-Menten type harvesting

<abstract><p>In this work, a new amensalism system with the nonlinear Michaelis-Menten type harvesting for the second species is studied. Firstly, we clarify topological types for all possible equilibria of the system. Then, the behaviors near infinity and the existence of closed orbits as well as saddle connections of the system are discussed via bifurcation analysis, and the global phase portraits of the model are also illustrated. Finally, for the sake of comparison, we further offer a new complete global dynamics of the model without harvesting. Numerical simulations show that the system with harvesting has far richer dynamics, like preserving the extinction of the first species or approaching the steady-state more slowly. Our research will provide useful information which may help us have a better understanding to the dynamic complexity of amensalism systems with harvesting effects.</p></abstract>

SINGULAR DIRECTIONS IN VEECH SURFACES

AbstractSingular directions in a Veech surface are shown to be exactly the directions of its saddle connections.

Systolic Geometry of Translation Surfaces

In this paper we investigate the systolic landscape of translation surfaces for fixed genus and fixed angles of their cone points. We furthermore study how the systoles of a translation surface relate to the systoles of its graph of saddle connections. This allows us to develop an algorithm to compute the systolic ratio of origamis in the stratum H ( 1 , 1 ) . We compute the maximal systolic ratio of all origamis in H ( 1 , 1 ) with up to 67 squares. These computations support a conjecture of Judge and Parlier about the maximal systolic ratio in H ( 1 , 1 ) .

Rigidity of the saddle connection complex

For a half-translation surface ( S , q ) $(S,q)$ , the associated saddle connection complex A ( S , q ) $\mathcal {A}(S,q)$ is the simplicial complex where vertices are the saddle connections on ( S , q ) $(S,q)$ , with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that any simplicial isomorphism ϕ : A ( S , q ) → A ( S ′ , q ′ ) $\phi \colon \mathcal {A}(S,q) \rightarrow \mathcal {A}(S^{\prime },q^{\prime })$ between saddle connection complexes is induced by an affine diffeomorphism F : ( S , q ) → ( S ′ , q ′ ) $F \colon (S,q) \rightarrow (S^{\prime },q^{\prime })$ . In particular, this shows that the saddle connection complex is a complete invariant of affine equivalence classes of half-translation surfaces. Throughout our proof, we develop several combinatorial criteria of independent interest for detecting various geometric objects on a half-translation surface.

Lower bounds for the number of limit cycles in a generalised Rayleigh–Liénard oscillator

In this paper a generalised Rayleigh–Liénard oscillator is consider and lower bounds for the number of limit cycles bifurcating from weak focus equilibria and saddle connections are provided. By assuming some open conditions on the parameters of the considered system the existence of up to twelve limit cycles is provided. More precisely, the approach consists in perform suitable changes in the sign of some specific parameters and apply Poincaré–Bendixson theorem for assure the existence of limit cycles. In particular, the algorithm for obtaining the limit cycles through the referred approach is explicitly exhibited. The main techniques applied in this study are the Lyapunov constants and the Melnikov method. The obtained results contemplate the simultaneity of limit cycles of small amplitude and medium amplitude, the former emerging from a weak focus equilibrium and the latter from homoclinic or heteroclinic saddle connections.

Experimental and numerical evaluation of the mechanical characteristics of semi‐rigid saddle connections

SummarySaddle connections are semi‐rigid connections that are widely used in Iran. Many existing buildings contain this type of connection. The present study conducted full‐scale experiments and used extensive numerical modeling to study the mechanical characteristics of saddle connections. The mechanical characteristics examined were the moment‐transfer mechanism, initial stiffness, yield moment, maximum moment, and fracture rotation. The configuration and dimensions of the experimental and numerical specimens were chosen to be similar to those of saddle connections in existing buildings. A parametric study was conducted to determine the factors affecting the mechanical characteristics of these connections. It was shown that the conventional method used to design saddle connections failed to predict the correct stress distribution. Empirical formulas were provided for predicting these mechanical characteristics and it was shown that the proposed expressions could predict the pertinent parameters accurately.

Continuation methods for principal foliations of embedded surfaces

<p style='text-indent:20px;'>Continuation methods are a well established tool for following equilibria and periodic orbits in dynamical systems as a parameter is varied. Properly formulated, they locate and classify bifurcations of these key components of phase portraits. Principal foliations of surfaces embedded in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula> resemble phase portraits of two dimensional vector fields, but they are not orientable. In the spirit of dynamical systems theory, Gutierrez and Sotomayor investigated qualitative geometric features that characterize structurally stable principal foliations and their bifurcations in one parameter families. This paper computes return maps and applies continuation methods to obtain new insight into bifurcations of principal foliations.</p><p style='text-indent:20px;'>Umbilics are the singularities of principal foliations and lines of curvature connecting umbilics are analogous to homoclinic and heteroclinic bifurcations of vector fields. Here, a continuation method tracks a periodic line of curvature in a family of surfaces that deforms an ellipsoid. One of the bifurcations of these periodic lines of curvature are connections between lemon umbilics. Differences between these bifurcations and analogous saddle connections in two dimensional vector fields are emphasized. A second case study tracks umbilics in a one parameter family of surfaces with continuation methods and locates their bifurcations using Taylor expansions in "Monge coordinates." Return maps that are generalized interval exchange maps of a circle are constructed for generic surfaces with no monstar umbilics.</p>

Improving the mechanical characteristics of semi-rigid saddle connections

There are many steel structures in Iran with semi-rigid saddle connections. It has been observed that, during a strong earthquake, this type of connection can experience a premature fracture, which will endanger the stability of the structure. The present study endeavored to improve the mechanical characteristics including the initial stiffness, yield moment, maximum moment, crack rotation, and load transfer mechanisms of the conventional saddle connections using retrofitting methods. An experimental study was conducted on six full-scale specimens to study the effect of the retrofitting methods on the stiffness, yield moment, and maximum moment of the connection and for the prevention of premature fracture. A numerical study including 88 high-fidelity finite element models was conducted to further evaluate two selected retrofitting methods. Using the generated database, predictive expressions have been proposed for calculating the initial stiffness, yield moment, and crack rotation. The proposed retrofitting methods were shown to improve the mechanical characteristics of conventional saddle connections.

Gaps of saddle connection directions for some branched covers of tori

AbstractWe compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit computations of gap distributions for non-lattice translation surfaces. We show that this distribution has support at zero and quadratic tail decay. We also construct examples of translation surfaces in any genus$d>1$that have the same gap distribution as the gap distribution of two identical tori glued along a slit. The second class we consider are twice-marked tori and saddle connections between distinct marked points with a specific orientation. These results can be interpreted as the gap distribution of slopes of affine lattices. We obtain our results by translating the question of gap distributions to a dynamical question of return times to a transversal under the horocycle flow on an appropriate moduli space.

Large-scale geometry of the saddle connection graph

We prove that the saddle connection graph associated to any half-translation surface is 4-hyperbolic and uniformly quasi-isometric to the regular countably infinite-valent tree. Consequently, the saddle connection graph is not quasi-isometrically rigid. We also characterise its Gromov boundary as the set of straight foliations with no saddle connections. In our arguments, we give a generalisation of the unicorn paths in the arc graph which may be of independent interest.

Translation covers of some triply periodic Platonic surfaces

We study translation covers of several triply periodic polyhedral surfaces that are intrinsically Platonic. We describe their affine symmetry groups and compute the quadratic asymptotics for counting saddle connections and cylinders, including the count of cylinders weighted by area. The mathematical study of triply periodic surfaces was initiated by Novikov, motivated by the study of electron transport. The surfaces we consider are of particular interest as they admit several different explicit geometric and algebraic descriptions, as described, for example, in the second author’s thesis.