Lyapunov 1-forms for flows

In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space $X$ admits a Lyapunov one-form $\omega$ lying in a prescribed \v{C}ech cohomology class $\xi\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $\xi$ to the chain recurrent set of $\Phi$. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions.Comment: 27 pages, 3 figures. This revised version incorporates a few minor improvement

Weak and strong fillability of higher dimensional contact manifolds

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five),while also being obstructed by all known manifestations of "overtwistedness". We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher-dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor edits. v3: exposition improved using referee's comments. Published by Invent. Mat

Counting solutions of perturbed harmonic map equations

In this paper we consider perturbed harmonic map equations for maps between closed Riemannian manifolds. In the case where the target manifold has negative sectional curvature we prove - among other results - that for a large class of semilinear and quasilinear perturbations, the perturbed harmonic map equations have solutions in any homotopy class of maps for which the Euler characteristic of the set of harmonic maps does not vanish. Under an additional condition, similar results hold in the case where the target manifold has nonpositive sectional curvature. The proofs are presented in an abstract setup suitable for generalizations to other situations

Flows with Lyapunov one-forms and a generalization of Farber's theorem on homoclinic cycles

We prove that if a flow on a compact manifold admits a Lyapunov one-form with small zero set Y, then there must exist a generalized homoclinic cycle, that is, a cyclically ordered chain of orbits outside Y such that for every consecutive pair the forward limit set of one and the backward limit set of the next are both contained in the same connected component of Y. The smallness of the zero set is measured in terms of a category-type invariant associated to the cohomology class of the form that was recently introduced by Farber (2002)

Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold

We show that for every closed Riemannian manifold X there exists a positive number¶ ε0>0 such that for all 00 such that for every metric space Y with Gromov-Hausdorff distance to X less than¶ δ the geometric ε -complex |Yε| is homotopy equivalent to X.¶ In particular, this gives a positive answer to a question of Hausmann

Smooth Lyapunov 1-forms

We find conditions which guarantee that a given flow O on a closed smooth manifold M admits a smooth Lyapunov 1-form u lying in a prescribed de Rham cohomology class £ 6 Hx (M; R). These conditions are formulated in terms of Schwartzman's asymptotic cycles A^($>) 6 //i(M;R) of the flow