Nonlocal asymptotic behavior of curves and leaves of laminations on universal coverings. Transl. from the Russian. (English) Zbl 1121.37001

Anosov, D. V.; Zhuzhoma, E. V.

Proceedings of the Steklov Institute of Mathematics 249. Moscow: Maik Nauka/Interperiodica. 221 p. (2005).

In this monograph, the authors study the topology of two-dimensional dynamical systems. More precisely, they study discrete systems, flows and foliations on surfaces equipped with metrics of constant nonpositive curvature. The exposition uses the classical methods of lifting the dynamical system to the universal covering, and it uses the dynamics induced on the circle at infinity in the case of negative curvature.
The authors review important results on the asymptotic behaviour of leaves and trajectories and their associated invariant manifolds. An example of the questions addressed is to know when equality of invariants of trajectories like asymptotic directions imply topological equivalence of the flows.
The topics discussed include the following: The Anosov theorem in the existence of asymptotic directions; Anosov’s theorems on the approximation of curves by semitrajectories; proof of the Weil theorem; Poincaré-Bendixson and generalizations; quasiminimal sets and laminations; limit sets; Anosov’s wild curve; deviation from coasymptotic geodesics; semitrajectories of flows; invariant manifolds; asymptotic behaviour of leaves; accessibility of points on the circle at infinity; classification results for flows, for irrational 2-webs, for minimal sets, for homeomorphisms of surfaces, for one-dimensional basic sets and for Cherry flows.
The monograph will be very useful for all researchers working on foliations and flows on surfaces.
The one article contained in this volume is a translation from the series “Tr. Mat. Inst. Steklova 249 (Russian)(2005)”.

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

37-02	Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37C85	Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37E35	Flows on surfaces
30F45	Conformal metrics (hyperbolic, Poincaré, distance functions)
30F50	Klein surfaces
57R30	Foliations in differential topology; geometric theory
53A04	Curves in Euclidean and related spaces
37D10	Invariant manifold theory for dynamical systems
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)

Keywords:

foliation; flow; asymptotic direction; Poincaré-Bendixson; minimal set; lamination; invariant set; trajectory.