Sectional-Anosov flows

We define sectional-Anosov flow as a vector field on a manifold, inwardly transverse to the boundary, whose maximal invariant set is sectional-hyperbolic (Metzger and Morales in Ergodic Theory Dyn Syst 28:1587–1597, 2008). We obtain properties of sectional-Anosov flows without null-homotopic periodic orbits on compact irreducible 3-manifolds including: incompressibility of transverse torus, non-existence of genus 0 transverse surfaces nor hyperbolic attractors nor hyperbolic repellers and sufficient conditions for the existence of singularities non-isolated in the nonwandering set. These generalize some known facts about Anosov flows.     

Sectional-Anosov flows on certain compact 3-manifolds

A sectional-Anosov flow is a flow for which the maximal invariant set is sectional-hyperbolic. A generalized 3-handlebody is a compact manifold which is built from a 3-disc attaching 0, 1, 2 and 3-handles at its boundary, one at a time, by attaching maps. We prove that there exist a class of orientable generalized 3-handlebodies supporting sectional-Anosov flows, moreover this class of manifolds is strictly large than the previous one studied in [14].     

An improved sectional-Anosov closing lemma

We prove that all nonwandering points of a sectional-Anosov flow on a compact 3-manifold can be approximated by periodic points or by points for which the omega-limit set is a singularity. This improves the closing lemma in Morales (Mich. Math. J. 56(1):29?C53, 2008). We also describe a sectional-Anosov flow for which the recurrent points are not dense in the nonwandering set.     

On the sensitivity of sectional-Anosov flows

We study sectional-Anosov flows on compact 3-manifolds for which the maximal invariant and nonwandering sets coincide. We prove that every vector field close to one of these flows is sensitive with respect to initial conditions.     

Unique ergodicity of horospheric foliations revisited

With each rational function on the Riemann sphere, Lyubich–Minsky construction (1997) associates an abstract topological space called the quotient hyperbolic lamination. The latter space carries the so-called vertical geodesic flow with Anosov property. Its unstable foliation is what we call the quotient horospheric lamination. We consider the case of hyperbolic rational function, and more generally, functions postcritically finite on the Julia set without parabolics, that do not belong to the following list of exceptions: powers, Chebyshev polynomials and Latt‘es examples. In this case the quotient horospheric lamination is known to be minimal, while restricted to the union of nonisolated hyperbolic leaves (Glutsyuk, 2007). In the present paper we prove its unique ergodicity. To this end, we introduce the so-called transversely contracting flows and homeomorphisms (on abstract compact metrizable topological spaces), which include the vertical geodesic flows under consideration and the usual Anosov flows and diffeomorphisms. We prove a version of our unique ergodicity result for the transversely contracting flows and homeomorphisms. Particular cases for Anosov flows and diffeomorphisms are given by classical results by Bowen, Marcus, Furstenberg, Margulis, et al. We give a new and purely geometric proof, which seems to be simpler than the classical ones (which use either Markov partitions, K-property, or harmonic analysis).     

C k-rigidity for hyperbolic flows II

In this note we prove the following result: Any conjugating homeomorphism between two geodesic flows for compact negatively curved compactC ^∞ surfaces is necessarilyC ^∞. This extends a result of Feldman and Ornstein. We also discuss some related results for hyperbolic flows and diffeomorphisms.     

A Note on Hyperbolic Flows in Sub-Riemannian Structures with Transverse Symmetries

The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrangian distribution) on a symplectic manifold. We consider the Hamiltonian flows of the curve of least action of natural mechanical systems in sub-Riemannian structures with symmetries. We give sufficient conditions for the reduced flows (after reduction of the first integrals induced from the symmetries) to be hyperbolic in terms of the reduced curvature and show new examples of Anosov flows.     

Nonuniform hyperbolicity for C1-generic diffeomorphisms

We study the ergodic theory of non-conservative C ¹-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C ¹-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C ¹-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.     

Invariant currents on limit sets

We relate the L ²-cohomology of a complete hyperbolic manifold to the invariant currents on its limit set. Received: January 18, 2000     

W-flows on Weyl manifolds and Gaussian thermostats

We introduce W-flows, by modifying the geodesic flow on a Weyl manifold, and show that they coincide with the isokinetic dynamics. We establish some connections between negative curvature of the Weyl structure and the hyperbolicity of W-flows, generalizing in dimension 2 the classical result of Anosov on Riemannian geodesic flows. In higher dimensions we establish only weaker hyperbolic properties. We extend the theory to billiard W-flows and introduce the Weyl counterparts of Sinai billiards. We obtain that the isokinetic Lorentz gas with the constant external field E and scatterers of radius r, studied by Chernov, Eyink, Lebowitz and Sinai, is uniformly hyperbolic, if only r|E|<1, and this condition is sharp.     

Hausdorff Dimension and Limits of Kleinian Groups

In this paper we prove that if M is a compact, hyperbolizable 3-manifold, which is not a handlebody, then the Hausdorff dimension of the limit set is continuous in the strong topology on the space of marked hyperbolic 3-manifolds homotopy equivalent to M. We similarly observe that for any compact hyperbolizable 3-manifold M (including a handlebody), the bottom of the spectrum of the Laplacian gives a continuous function in the strong topology on the space of topologically tame hyperbolic 3-manifolds homotopy equivalent to M. Submitted: January 1998.     

Hyperbolic Structure for the Quadratic Map

We consider a quadratic map of the real plane and suggest a construction of a hyperbolic structure on a certain noncompact set. Bibliography: 1 title.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 266–272.     

Bergman Space Structure,Commutative Algebras of Toeplitz Operators,and Hyperbolic Geometry

We exhibit a surprising but natural connection among the Bergman space structure, commutative algebras of Toeplitz operators and pencils of hyperbolic straight lines. The commutative C*-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines the set of symbols consisting of functions which are constant on corresponding cycles, the orthogonal trajectories to lines forming a pencil. It turns out that the C*-algebra generated by Toeplitz operators with this class of symbols is commutative. Submitted: January 15, 2001?Revised: February 7, 2002     

Random C 0 homeomorphism perturbations of hyperbolic sets

In this note we consider random C ⁰ homeomorphism perturbations of a hyperbolic set of a C ¹ diffeomorphism. We show that the hyperbolic set is semi-stable under such perturbations, in particular, the topological entropy will not decrease under such perturbations.     

Numerical study of the transverse stability of the Peregrine solution

We generalize a previously published numerical approach for the one-dimensional (1D) nonlinear Schrödinger (NLS) equation based on a multidomain spectral method on the whole real line in two ways: first, a fully explicit fourth-order method for the time integration, based on a splitting scheme and an implicit Runge-Kutta method for the linear part, is presented. Second, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the 1D NLS equation and thus a y-independent solution to the 2D NLS. It is shown that the Peregine solution is unstable agains all standard perturbations, and that some perturbations can even lead to a blow-up for the elliptic NLS equation.     

The Isomorphism Problem for All Hyperbolic Groups

We give a solution to Dehn’s isomorphism problem for the class of all hyperbolic groups, possibly with torsion. We also prove a relative version for groups with peripheral structures. As a corollary, we give a uniform solution to Whitehead’s problem asking whether two tuples of elements of a hyperbolic group G are in the same orbit under the action of Aut(G). We also get an algorithm computing a generating set of the group of automorphisms of a hyperbolic group preserving a peripheral structure.     

Higher dimensional extensions of substitutions and their dual maps

Given a substitution σ ond letters, we define itsk-dimensional extension,E _k (σ), for 0≤k≤d. Thek-dimensional extension acts on the set ofk-dimensional faces of unit cubes inR ^d with integer vertices. The extensions of a substitution satisfy a commutation relation with the natural boundary operator: the boundary of the image is the image of the boundary. We say that a substitution is unimodular (resp. hyperbolic) if the matrix associated to the substitution by abelianization is unimodular (resp. hyperbolic). In the case where the substitution is unimodular, we also define dual substitutions which satisfy a similar coboundary condition. We use these constructions to build self-similar sets on the expanding and contracting space for an hyperbolic substitution.     

The structure of branching in Anosov flows of 3-manifolds

In this article we study the topology of Anosov flows in 3-manifolds. Specifically we consider the lifts to the universal cover of the stable and unstable foliations and analyze the leaf spaces of these foliations. We completely determine the structure of the non Hausdorff points in these leaf spaces. There are many consequences: (1) when the leaf spaces are non Hausdorff, there are closed orbits in the manifold which are freely homotopic, (2) suspension Anosov flows are, up to topological conjugacy, the only Anosov flows without free homotopies between closed orbits, (3) when there are infinitely many stable leaves (in the universal cover) which are non separated from each other, then we produce a torus in the manifold which is transverse to the Anosov flow and therefore is incompressible, (4) we produce non Hausdorff examples in hyperbolic manifolds and derive important properties of the limit sets of the stable/unstable leaves in the universal cover. Received: March 13, 1997     

Local maximality of hyperbolic sets

Two properties of a hyperbolic set F are discussed: its local maximality and the property that, in any neighborhood U ⊃ F, there exists a locally maximal set F′ that contains F (we suggest calling the latter property local premaximality). Although both these properties of the set F are related to the behavior of trajectories outside F, it turns out that, in the class of hyperbolic sets, the presence or absence of these properties is determined by the interior dynamics on F.