Nonexistence of Bonatti-Langevin examples of Anosov flows on closed four manifolds

Bonatti and Langevin constructed an Anosov flow on a closed 3-manifold with a transverse torus intersecting all orbits except one [C. Bonatti, R. Langevin, Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension, Ergodic Theory Dynam. Systems 14 (4) (1994), 633-643]. We shall prove that these flows cannot be constructed on closed 4-manifolds. More precisely, there are no Anosov flows on closed 4-manifolds with a closed, incompressible, transverse submanifold intersecting all orbits except finitely many closed ones. The proof relies on the analysis of the trace of the weak invariant foliations of the flow on the transverse submanifold.     

On Reeb components of invariant foliations of projectively Anosov flows

We show that if a C² codimension one foliation on a three-dimensional manifold has a Reeb component and is invariant under a projectively Anosov flow, then it must be a Reeb foliation on S²×S¹.     

A λ-lemma for foliations

We show that a C⁰ codimension one foliation with C¹ leaves F of a closed manifold is minimal if there are a foliation G transverse to F, and a diffeomorphism f preserving both foliations, such that every leaf of F intersects every leaf of G and f expands G. We use this result to study of Anosov actions on closed manifolds.     

Flat Manifolds With Prescribed First Betti Number Admitting Anosov Diffeomorphisms

 We show that from dimension six onwards (but not in lower dimensions), there are in each dimension flat manifolds with first Betti number equal to zero admitting Anosov diffeomorphisms. On the other hand, it is known that no flat manifolds with first Betti number equal to one support Anosov diffeomorphisms. For each integer k > 1 however, we prove that there is an n-dimensional flat manifold M with first Betti number equal to k carrying an Anosov diffeomorphism if and only if M is a k-torus or n is greater than or equal to k + 2. (Received 5 October 2000; in revised form 9 March 2001)     

Zeta-functions for expanding maps and Anosov flows

Given a real-analytic expanding endomorphism of a compact manifoldM, a meromorphic zeta function is defined on the complex-valued real-analytic functions onM. A zeta function for Anosov flows is shown to be meromorphic if the flow and its stable-unstable foliations are real-analytic.     

Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows

We show that certain mechanical systems, including a geodesic flow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy.The assumptions we make in the case of geodesic flows are:

(a)

The metric and the external perturbation are smooth enough.

(b)

The geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection.

(c)

The frequency of the external perturbation is Diophantine.

(d)

The external potential satisfies a generic condition depending on the periodic orbit considered in (b).

The assumptions on the metric are C² open and are known to be dense on many manifolds. The assumptions on the potential fail only in infinite codimension spaces of potentials.The proof is based on geometric considerations of invariant manifolds and their intersections. The main tools include the scattering map of normally hyperbolic invariant manifolds, as well as standard perturbation theories (averaging, KAM and Melnikov techniques).We do not need to assume that the metric is Riemannian and we obtain results for Finsler or Lorentz metrics. Indeed, there is a formulation for Hamiltonian systems satisfying scaling hypotheses. We do not need to make assumptions on the global topology of the manifold nor on its dimension.     

Law of the iterated logarithm for transitiveC 2 Anosov flows and semiflows over maps of the interval

It is proved that a functional law of the iterated logarithm is valid for transitiveC ² Anosov flows on compact Riemannian manifolds when the observable belongs to a certain class of real-valued Hölder functions. The result is equally valid for semiflows over piecewise expanding interval maps that are similar to the Williams' Lorenz-attractor semiflows. Furthermore the observables need only be real-valued Hölder for these semiflows.     

Cocycles over interval exchange transformations and multivalued Hamiltonian flows

We consider interval exchange transformations of periodic type and construct different classes of ergodic cocycles of dimension?1 over this special class of IETs. Then using Poincaré sections we apply this construction to obtain the recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.     

On the approximation of time one maps of Anosov flows by Axiom A diffeomorphisms

We prove that if 𝒻₁ is the time one map of a transitive and codimension one Anosov flow φ and it is C ¹-approximated by Axiom A diffeomorphisms satisfying a property called P, then the flow is topologically conjugated to the suspension of a codimension one Anosov diffeomorphism. A diffeomorphism 𝒻 satisfies property P if for every periodic point in M the number of periodic points in a fundamental domain of its central manifold is constant. Received: 15 March 2001     

Homoclinic tangencies versus uniform hyperbolicity for conservative 3-flows

We prove that a volume-preserving three-dimensional flow can be C¹-approximated by a volume-preserving Anosov flow or else by another volume-preserving flow exhibiting a homoclinic tangency. This proves the conjecture of Palis for conservative 3-flows and with respect to the C¹-topology.     

Some ergodic and rigidity properties of discrete Heisenberg group actions

The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group \(\mathcal{H}\). We establish the decomposition of the tangent space of any C^∞ compact Riemannian manifold M for Lyapunov exponents, and show that all Lyapunov exponents for the central elements are zero. We obtain that if an \(\mathcal{H}\) action contains an Anosov element, then under certain conditions on the eigenvalues of this element, the action of each central element is of finite order. In particular, there is no faithful codimension one Anosov Heisenberg group action on any compact manifold, and there is no faithful codimension two Anosov Heisenberg group action on tori. In addition, we show smooth local rigidity for higher rank ergodic \(\mathcal{H}\) actions by toral automorphisms, using a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.     

Topological stability for conservative systems

We prove that the C¹ interior of the set of all topologically stable C¹ incompressible flows is contained in the set of Anosov incompressible flows. Moreover, we obtain an analogous result for the discrete-time case.     

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).     

Generic robustness of spectral decompositions

We prove that given a compact n-dimensional boundaryless manifold M, n?2, there exists a residual subset R of Diff¹(M) such that if f∈R admits a spectral decomposition (i.e., the non-wandering set admits a partition into a finite number of transitive compact sets), then this spectral decomposition is robust in a generic sense (tame behavior). This implies a C¹-generic trichotomy that generalizes some aspects of a two-dimensional theorem of Mañé [Topology 17 (1978) 386-396].Lastly, Palis [Astérisque 261 (2000) 335-347] has conjectured that densely in Diff^k(M) diffeomorphisms either are hyperbolic or exhibit homoclinic bifurcations. We use the aforementioned results to prove this conjecture in a large open region of Diff¹(M).     

Unique ergodicity of the horocycle flow: Variable negative curvature case

H. Furstenberg showed that horocycle flows on compact manifolds of constant negative curvature are uniquely ergodic. This paper generalizes his result to the case of variable negative curvature, in the more general context of flows whose orbits are the unstable manifolds of certain Anosov flows.     

Torsion and conformally Anosov flows in contact Riemannian geometry

We prove that on a compact (non Sasakian) contact metric 3-manifold with critical metric for the Chern-Hamilton functional, the characteristic vector field ξ is conformally Anosov and there exists a smooth curve in the contact distribution of conformally Anosov flows. As a consequence, we show that negativity of the ξ-sectional curvature is not a necessary condition for conformal Anosovicity of ξ (this completes a result of [4]). Moreover, we study contact metric 3-manifolds with constant ξ-sectional curvature and, in particular, correct a result of [13].     

A New Proof of the Existence of Embedded Surfaces with Anosov Geodesic Flow

We give a new proof of the existence of compact surfaces embedded in ?³ with Anosov geodesic flows. This proof starts with a noncompact model surface whose geodesic flow is shown to be Anosov using a uniformly strictly invariant cone condition. Using a sequence of explicit maps based on the standard torus embedding, we produce compact embedded surfaces that can be seen as small perturbations of the Anosov model system and hence are themselves Anosov.     

Characterizing omega-limit sets which are closed orbits

Let X be a vector field in a compact n-manifold M, n?2. Given Σ⊂M we say that q∈M satisfies (P)_Σ if the closure of the positive orbit of X through q does not intersect Σ, but, however, there is an open interval I with q as a boundary point such that every positive orbit through I intersects Σ. Among those q having saddle-type hyperbolic omega-limit set ω(q) the ones with ω(q) being a closed orbit satisfy (P)_Σ for some closed subset Σ. The converse is true for n=2 but not for n?4. Here we prove the converse for n=3. Moreover, we prove for n=3 that if ω(q) is a singular-hyperbolic set [C. Morales, M. Pacifico, E. Pujals, On C¹ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris Sér. I 26 (1998) 81-86], [C. Morales, M. Pacifico, E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2) 160 (2) (2004) 375-432], then ω(q) is a closed orbit if and only if q satisfies (P)_Σ for some Σ closed. This result improves [S. Bautista, Sobre conjuntos hiperbólicos-singulares (On singular-hyperbolic sets), thesis Uiversidade Federal do Rio de Janeiro, 2005 (in Portuguese)] and [C. Morales, M. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001) 359-378].     

Markov structures and decay of correlations for non-uniformly expanding dynamical systems

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure. Moreover, the decay of the return time function can be controlled in terms of the time generic points need to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem.     

Boundary of Anosov dynamics and evolution equations for surfaces

We show that a compact surface of genus greater than one, without focal points and a finite number of bubbles (“good” shaped regions of positive curvature) is in the closure of Anosov metrics. Compact surfaces of nonpositive curvature and genus greater than one are in the closure of Anosov metrics, by Hamilton's work about the Ricci flow. We generalize this fact to the above surfaces without focal points admitting regions of positive curvature using a “magnetic” version of the Ricci flow, the so‐called Ricci Yang‐Mills flow.