Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle

We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are $C^r$ open and there exists a $C^r$ open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.     

Leaf closures of Riemannian foliations: A survey on topological and geometric aspects of Killing foliations

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply-connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, sheaves, holonomy and basic cohomology. We then review Molino’s structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations, including the recent proof of Molino’s conjecture, and discuss singular Killing foliations.     

On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces

Given a rational homology classh in a two dimensional torusT ², we show that the set of Riemannian metrics inT ² with no geodesic foliations having rotation numberh isC ^k dense for everyk N. We also show that, generically in theC ² topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.Partially supported by CNPq-GMD, FAPERJ, and the University of Freiburg.     

Hyperbolicity, heterodimensional cycles and Lyapunov exponents for partially hyperbolic dynamics

We prove a dichotomy of C² partially hyperbolic sets with one-dimensional center direction admitting no zero Lyapunov exponents, either hyperbolicity over the supports of ergodic measures or approximation by a heterodimensional cycle. This provides a partial result to the C¹ Palis Conjecture that claims a dichotomy, hyperbolicity or homoclinic bifurcations in a dense subset of the space of C¹ diffeomorphisms. Moreover, a theorem of Ma?é applied in the proof is modified to have an additional property concerning the Hausdorff distance between a periodic orbit and the support of a hyperbolic ergodic measure.     

Surface billiards and nonvanishing Lyapunov exponents

We study in this paper the billiards on surfaces with mix-valued Gaussian curvature and the condition which gives nonvanishing Lyapunov exponents of the system. We introduce a criterion upon which a small perturbation of the surface will also produce a system with positive Lyapunov exponents. Some examples of such surfaces are given in this article.     

Parallel focal structure and singular Riemannian foliations

We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.

     

Singular riemannian foliations with sections,transnormal maps and basic forms

A singular riemannian foliation on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold Σ that meets every leaf of orthogonally and whose dimension is the codimension of the regular leaves of . We prove that the algebra of basic forms of M relative to is isomorphic to the algebra of those differential forms on Σ that are invariant under the generalized Weyl pseudogroup of Σ. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos. Marcos M. Alexandrino and Claudio Gorodski have been partially supported by FAPESP and CNPq.     

Hyperbolicity, transversality and analytic first integrals

Let A be a (normally) hyperbolic compact invariant manifold of an analytic diffeomorphism f of an analytic manifold M. We assume that the stable and unstable manifold of A intersect transversally (in an admissible way), the dynamics on A is ergodic and the modulus of the eigenvalues associated to the stable and unstable manifold, respectively, satisfy a non-resonance condition. In the case where A is a point or a torus, we prove that the discrete dynamical system associated to f does not admit an analytic first integral. The proof is based on a triviality lemma, which is of combinatorial nature, and a geometrical lemma. The same techniques, allow us to prove analytic non-integrability of Hamiltonian systems having Arnold diffusion. In particular, using results of Xia, we prove analytic non-integrability of the elliptic restricted three-body problem, as well as the planar three-body problem.     

Lyapunov exponents,bifurcation currents and laminations in bifurcation loci

Bifurcation loci in the moduli space of degree d rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period n and multiplier 0 or e ^iθ. Using potential-theoretic arguments, we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current. To this purpose we first establish approximation formulas for the Lyapunov function. In degree d = 2, this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin exists.     

Decomposition of stochastic flows and Lyapunov exponents

Let φ _t be the stochastic flow of a stochastic differential equation on a compact Riemannian manifold M. Fix a point m∈M and an orthonormal frame u at m, we will show that there is a unique decomposition φ _t = ξ _t ψ _t such that ξ _t is isometric, ψ _t fixes m and Dψ _t (u) = us _t , where s _t is an upper triangular matrix. We will also establish some convergence properties in connection with the Lyapunov exponents and the decomposition Dφ _t (u) = u _t s _t with u _t being an orthonormal frame. As an application, we can show that ψ_t preserves the directions in which the tangent vectors at m are dilated at fixed exponential rates. Received: 19 November 1998 / Revised version: 1 October 1999 / Published online: 14 June 2000     

A general approach to index theorems for holomorphic maps and foliations

Let M be a smooth complex manifold, and S(⊂ M) be a compact irreducible subvariety with dim _C S > 0. Let be given either a holomorphic map f : M → M with f _|S  = id _S , f ≠ id _M , or a holomorphic foliation on M: we describe an approach that can be applied to both map and foliation in order to obtain index theorems. Partially supported by GNSAGA, Centro de Giorgi, M.U.R.S.T.     

Asymptotic phase and invariant foliations near periodic orbits

The paper deals with asymptotic phase and invariant foliations near periodic orbits, extending for two-dimensional smooth vector fields results that have been obtained by Chicone and Liu (2004). The problem of the existence of asymptotic phase is completely solved for analytic vector fields and is reduced to a problem of infinite codimension for systems. Moreover it is proven that whenever asymptotic phase occurs, or in other words, when the periodic orbit is isochronous, then there also exists a foliation, with leaves transversally cutting the periodic orbit and invariant under the flow of the vector field. The paper also provides some results in three dimensions.

     

Lyapunov exponents and control sets near singular points

We consider control affine systems of the form

     

Almost sure and moment Lyapunov exponents for a stochastic beam equation

The transverse vibrations of an Euler-Bernoulli beam with axial tension P and axial white noise forcing are given by

     

Lyapunov exponents and asymptotic dynamics in random Kolmogorov models

The current paper is devoted to the investigation of asymptotic dynamics in random Kolmogorov models. Applying the theory of principal Lyapunov exponents and the principal spectrum developed in the authors previous papers together with the concept of part metric it provides conditions for the existence of a globally attracting positive random equilibrium, the existence of a globally attracting uniformly positive random equilibrium, and the extinction in random Kolmogorov models. These results are an important complement to the existing ones.     

Hyperbolicity versus partial-hyperbolicity and the transversality-torsion phenomenon

In this paper, we describe a process to create hyperbolicity in the neighbourhood of a homoclinic orbit to a partially hyperbolic torus for three degrees of freedom Hamiltonian systems: the transversality-torsion phenomenon.     

Accessibility and homology bounded strong unstable foliation for Anosov diffeomorphisms on 3-torus

We give three equivalent conditions for non-accessibility of an Anosov diffeomorphism on the 3-torus with a partially hyperbolic splitting. Since accessibility is an open property, this gives a negative answer to Hammerlindl’s question about homology boundedness of strong unstable foliation.     

A game with divisors and absolute differences of exponents

In this work, we discuss a number game that develops in a manner similar to that on which Gilbreath's conjecture on iterated absolute differences between consecutive primes is formulated. In our case the action occurs at the exponent level and there, the evolution is reminiscent of that in a final Ducci game. We present features of the whole field of the game created by the successive generations, prove an analogue of Gilbreath's conjecture and raise some open questions.     

Green bundles,Lyapunov exponents and regularity along the supports of the minimizing measures

In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as:

•

the Lyapunov exponents of minimizing measures;