Related papers: Lyapunov graphs of nonsingular Smale flows on $S^{…
In this paper, we use abstract Lyapunov graphs as a combinatorial tool to obtain a complete classification of Smale flows on $\mathbb{S}^2\times\mathbb{S}^1$. This classification gives necessary and sufficient conditions that must be…
In this paper, we first discuss the regular level set of a nonsingular Smale flow (NSF) on a 3-manifold. The main result about this topic is that a 3-manifold $M$ admits an NSF flow which has a regular level set homeomorphic to $(n+1)T^{2}$…
In this paper, we first develope the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For…
In this paper, we first discuss some connections between template theory and the description of basic sets of Smale flows on 3-manifolds due to F. B\'eguin and C. Bonatti. The main tools we use are symbolic dynamics, template moves and some…
In this work we address the realizability of a Lyapunov graph labeled with GS singularities, namely regular, cone, Whitney, double crossing and triple crossing singularities, as continuous flow on a singular closed $2$-manifold…
We find conditions which guarantee that a given flow on a closed smooth manifold admits a smooth Lyapunov one-form lying in a prescribed de Rham cohomology class. These conditions are formulated in terms of Schwartzman's asymptotic cycles…
We give necessary and sufficient conditions for the existence of smooth Lyapunov 1-forms for the flow of a smooth vector field in terms of the behavior of certain locally finite invariant measures. The main statement generalizes a result of…
The goal of the present work is to compute explicitely the correlation spectrum of a Morse-Smale flow in terms of the Lyapunov exponents of the Morse--Smale flow, the topology of the flow around periodic orbits and the monodromy of some…
We introduce and analyze a notion of smooth Lyapunov 1-form for flows generated by vector fields on orbifolds. Using asymptotic cycles and chain-recurrent sets, we establish topological conditions that guarantee the existence of a Lyapunov…
We study $2$-dimensional unit vector flows on graphs, that is, nowhere-zero flows that assign to each oriented edge a unit vector in $\mathbb R^{3}$. We give a new geometric characterization of $\mathbb S^{2}$-flows on cubic graphs. We also…
A Smale flow is a structurally stable flow with one dimensional invariant sets. We use information from homology and template theory to construct, visualize and in some cases, classify, nonsingular Smale flows in the 3-sphere.
We construct a complete invariant for non-wandering surface flows with finitely many singular points but without locally dense orbits. Precisely, we show that a flow $v$ with finitely many singular points on a compact connected surface $S$…
This is the first article in a series that aims at classifying partial sections of flows, that is a general family of transverse surfaces. In this part, we deal with the dynamical aspect of the question. Given a flow on a compact manifold…
We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize…
We prove the existence of a discrete correlation spectrum for Morse-Smale flows acting on smooth forms on a compact manifold. This is done by constructing spaces of currents with anisotropic Sobolev regularity on which the Lie derivative…
We give the complete classification of regular projectively Anosov flows on closed three-dimensional manifolds. More precisely, we show that such a flow must be either an Anosov flow or decomposed into a finite union of $T^2 \times…
We study time-changes of unipotent flows on finite volume quotients of semisimple linear groups, generalising previous work by Ratner on time-changes of horocycle flows. Any measurable isomorphism between time-changes of unipotent flows…
We give complete classification of C^2-regular and non-degenerate projectively Anosov flows on three dimensional manifolds. More precisely, we prove that such a flow on a connected manifold must be either an Anosov flow or represented as a…
Using results relating taut foliations and pseudo-Anosov flows, we find cusped hyperbolic 3-manifolds which are not the non-singular part of a pseudo-Anosov flow. In particular, we find the first examples of cusped hyperbolic 3-manifolds…
The embedded template is a geometric tool in dynamics being used to model knots and links as periodic orbits of $3$-dimensional flows. We prove that for an embedded template in $S^3$ with fixed homeomorphism type, its boundary as a…