arXiv subjects

J. Latschev

Publications and source records attributed to J. Latschev.

3 recordsLinked to original sources

arXiv2005

Quantitative symplectic geometry

While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. In this paper we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. We also describe several new relations between certain symplectic capacities on ellipsoids and polydiscs. Throughout the discussion we mention many open problems.

math.SG↗

arXiv2003

Smooth Lyapunov 1-forms

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 of the flow.

math.DS↗

arXiv2003

Lyapunov 1-forms for flows

In this paper we find conditions which guarantee that a given flow $Φ$ on a compact metric space $X$ admits a Lyapunov one-form $ω$ lying in a prescribed Čech cohomology class $ξ\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $ξ$ to the chain recurrent set of $Φ$. 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.

math.DS↗