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 $c R^2$ 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 $L^2$. 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 $c R^2$ where $c$ depends in this case on the given lattice surface.
We give effective estimates for the number of saddle connections on a translation surface that have length $\leq L$ and are in a prescribed homology class modulo $q$. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum.
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.
We consider an open and dense subset of the set of all connections (heteroclinic orbits) between two saddles of a vector field in dimension three. We give a complete classification up to topological equivalence of the dynamics in a neighbourhood of a connection in this subset.
An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian and two non-symmetric heteroclinic orbits permuted by the involution. This is a codimension one structure and therefore it can be met generally in one-parameter families of reversible Hamiltonian systems. There exist two possible types of such connections in dependence on how the involution acts near the equilibrium. We prove a series…
In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interaction with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection allowed to cross or slide along the discontinuity. Even the simplest case, that of connection to a …
We prove that the asymptotic number of pairs of saddle connections with length smaller than $L$ with bounded virtual area is quadratic for almost every translation surface with respect to any ergodic $SL(2,\mathbb{R})$-invariant measure. A key tool of the proof is that Siegel-Veech transforms of bounded functions with compact supports are in $L^{2+κ}$ for every $SL(2,\mathbb{R})$-invariant measure.
We prove an effective version of a celebrated result of Eskin and Masur: for any affine invariant manifold of translation surfaces, almost every translation surface has quadratic growth for the saddle connection holonomy vectors, with an effective bound of the error. We also provide effective versions of counting in sectors and in ellipses.
Asymptotic quadratic growth rates of saddle connections and families of periodic cylinders on translation tori with n marked points are studied. For any marking the existence of limits of the quadratic growth rate is shown using elementary methods (not Ratners theorem). We study the growth rate limit as function of the marking. We give precise formulas for this function in the case of two marked points and describe the sets where the growth function is maximal and continuous in any case. For rational two markings we calculate the index of the Veech group in SL(2,Z) using two different ways.
Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential $ω$ on a Riemann surface of genus $g \geq 2$ the smallest gap between saddle connection directions of length at most a fixed length decays faster than quadratically in the length. We also characterize the exceptional set: the decay rate is not faster than quadratic if and only if $ω$ is a lattice surface.
A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. The similar phenomenon is valid for the families of parallel closed geodesics. We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geo…
We 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 0 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 …
Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces may contain trajectories accumulating on a limit cycle. Such a limit cycle is called hyperbolic because it induces a nontrivial homothety. It has been conjectured that a dilation surface with no actual hyperbolic closed geodesic is in fact a translation surface. Assuming that a dilation surface contains a horizon saddle connection, we prove that the directions…
Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine interval exchange maps. However, though the $SL(2,\mathbb{R})$-action is ergodic in connected components of strata of translation surfaces, there may be mutually disjoint $SL(2,\mathbb{R})$-invariant open subsets in components of dilation surfaces. A first distinction is between triangulable and non-triangulable dilation surfaces. For non-triangulable surfaces, th…
We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincaré section for the horocycle flow on $SL(2,\mathbb{R})/SL(X,ω)$ associated to an arbitrary Veech surface $(X, ω)$. As a corollary, we show that the associated gap distribution is piecewise real analytic.
In this paper, a modified van der Pol oscillator equation is considered which appears in several heart action models. We study its global dynamics and verify many interesting bifurcations such as a Hopf bifurcation, a heteroclinic saddle connection, and a homoclinic saddle connection. Some of these bifurcations are detected by using Conley index methods. We demonstrate how the study of connection matrices and transition matrices shows how to select interesting parameter values for simulations.
Effective shadowing of transition maps for a Hamiltonian saddle-center-center equilibrium
Jet Propulsion Laboratory, California Institute of Technology（加州理工学院喷气推进实验室）；Laboratory of Geometry and Dynamical Systems and IMTech, UPC（UPC几何与动力系统实验室和IMTech）；Department of Mathematical Sciences, Yeshiva University（叶史亚大学数学科学系）
We explore sequential escape behaviour of coupled bistable systems under the influence of stochastic perturbations. We consider transient escapes from a marginally stable "quiescent" equilibrium to a more stable "active" equilibrium. The presence of coupling introduces dependence between the escape processes: for diffusive coupling there is a strongly coupled limit (fast domino regime) where the escapes are strongly synchronised while for intermediate coupling (slow domino regime) without partially escaped stable states, there is still a delayed effect. These regimes can be associated with bif…
We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments.
Smooth planar vector fields containing two hyperbolic saddles may possess contours formed by heteroclinic connections between these saddles. We present an overview of the bifurcations of these contours based on papers by J.W. Reyn and A.V. Dukov. Additionally, two new explicit polynomial systems containing such contours are derived, which are studied using the bifurcation software MatCont and are shown to exhibit the theoretically predicted phenomena, including series of heteroclinic connections.
We show that the number of square-tiled surfaces of genus $g$, with $n$ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $L$ squares, is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.
Explicit formulae are given for the saddle connection for an integrable family of standard maps studied by Suris. A generalization of Melnikov's method shows that, upon perturbation, this connection is destroyed. We give explicit formula for the first order approximation of the area of the lobes of the resultant turnstile. It is shown that the lobe area is exponentially small in the limit when the Suris map approaches the trivial twist map.
In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles (corresponding to meromorphic differentials). The Cantor-Bendixson rank of their set of directions is a measure of descriptive set-theoretic complexity. Drawing on a previous work of David Aulicino, we prove a sharp upper bound that depends only on the genus of the underlying topological surface. The proof uses a new geometric lemma stating that in a sequence of three …
We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincaré and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect …
