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     

Robustly transitive actions of ℝ<Superscript>2</Superscript> on compact three manifolds

In this paper, we define robust transitivity for actions of ℝ² on closed connected orientable manifolds. We prove that if the ambient manifold is three dimensional and the dense orbit of a robustly transitive action is not planar, then the action is defined by an Anosov flow, i.e. its orbits coincide with the orbits of an Anosov flow.     

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.     

Nonexistence of codimension one Anosov flows on compact manifolds with boundary

A flow is Anosov if it exhibits contracting and expanding directions forming with the flow a continuous tangent bundle decomposition. An Anosov flow is codimension one if its contracting or expanding direction is one-dimensional. Examples of codimension one Anosov flows on compact boundaryless manifolds can be exhibited in any dimension ?3. In this paper, we prove that there are no codimension one Anosov flows on compact manifolds with boundary. The proof uses an extension to flows of some results in Hirsch [On Invariant Subsets of Hyperbolic Sets, Essays on Topology and Related Topics, Memoires dédiés à Georges de Rham, 1970, pp. 126-135] related to Question 10(b) in Palis and Pugh [Fifty problems in dynamical systems, in: J. Palis, C.C. Pugh (Eds.), Dynamical Systems-Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E.C. Zeeman on his fiftieth birthday), Lecture Notes in Mathematics, vol. 468, Springer, Berlin, 1975, pp. 345-353].     

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.     

Flow equivalence,hyperbolic systems and a new zeta function for flows

We analyze the dynamics of diffeomorphisms in terms of their suspension flows. For many Axion A diffeomorphisms we find simplest representatives in their flow equivalence class and so reduce flow equivalence to conjugacy. The zeta functions of maps in a flow equivalence class are correlated with a zeta function ζ _H for their suspended flow. This zeta function is defined for any flow with only finitely many closed orbits in each homology class, and is proven rational for Axiom A flows. The flow equivalence of Anosov diffeomorphisms is used to relate the spectrum of the induced map on first homology to the existence of fixed points. For Morse-Smale maps, we extend a result of Asimov on the geometric index. Partially supported by MCS 76-08795.     

Pseudo-Anosov Flows and Incompressible Tori

We study incompressible tori in 3-manifolds supporting pseudo-Anosov flows and more generally ZZ subgroups of the fundamental group of such a manifold. If no element in this subgroup can be represented by a closed orbit of the pseudo-Anosov flow, we prove that the flow is topologically conjugate to a suspension of an Anosov diffeomorphism of the torus. In particular it is non singular and is an Anosov flow. It follows that either a pseudo-Anosov flow is topologically conjugate to a suspension Anosov flow, or any immersed incompressible torus can be realized as a free homotopy from a closed orbit of the flow to itself. The key tool is an analysis of group actions on non-Hausdorff trees, also known as R-order trees – we produce an invariant axis in the free action case. An application of these results is the following: suppose the manifold has an R-covered foliation transverse to a pseudo-Anosov flow. If the flow is not an R-covered Anosov flow, then it follows that the manifold is atoroidal.     

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.     

Prime orbit theorems with multi-dimensional constraints for axiom a flows

In this paper we give a new proof of an asymptotic formula for the number of closed orbits of an Axiom A flow subject to certain constraints due to S. P. Lalley. We consider finite group extensions and, for transitive Anosov flows, give an application to homology. We also discuss asymptotics for closed orbits in a fixed homology class, extending a result of Katsuda and Sunada.Supported by S.E.R.C. grant no. 88001623.     

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.     

An example on movable approximations of a minimal set in a continuous flow

In the present paper the study of flows on n-manifolds in particular in dimension three, e.g., R³, is motivated by the following question. Let A be a compact invariant set in a flow on X. Does every neighbourhood of A contain a movable invariant set M containing A? It is known that a stable solenoid in a flow on a 3-manifold has approximating periodic orbits in each of its neighbourhoods. The solenoid with the approximating orbits form a movable set, although the solenoid is not movable. Not many such examples are known. The main part of the paper consists of constructing an example of a set in R³ that is not stable, is not a solenoid, and is approximated by Denjoy-like invariant sets instead of periodic orbits. As in the case of a solenoid, the constructed set is an inverse limit of its approximating sets. This gives a partial answer to the above question.     

On invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture

We show that any topologically transitive codimension-one Anosov flow on a closed manifold is topologically equivalent to a smooth Anosov flow that preserves a smooth volume. By a classical theorem due to Verjovsky, any higher-dimensional codimension-one Anosov flow is topologically transitive. Recently, Simić showed that any higher-dimensional codimension-one Anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an Anosov diffeomorphism. Therefore, our result gives a complete classification of codimension-one Anosov flows up to topological equivalence in higher dimensions.     

Transitive partially hyperbolic diffeomorphisms on 3-manifolds

The known examples of transitive partially hyperbolic diffeomorphisms on 3-manifolds belong to 3 basic classes: perturbations of skew products over an Anosov map of T², perturbations of the time one map of a transitive Anosov flow, and certain derived from Anosov diffeomorphisms of the torus T³. In this work we characterize the two first types by a local hypothesis associated to one closed periodic curve.     

On the cardinality of the θ-closed hull of sets

The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all closed neighborhoods of a point intersect C, this point is in C.     

Geometric Anosov flows of dimension 5

We show that for a smooth Anosov flow on a closed five dimensional manifold, if it has C^∞ Anosov splitting and preserves a C^∞ pseudo-Riemannian metric, then up to a special time change and finite covers, it is C^∞ flow equivalent either to the suspension of a symplectic hyperbolic automorphism of $\text{T}{}^4$, or to the geodesic flow on a three dimensional hyperbolic manifold. To cite this article: Y. Fang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the realization of Riemannian symmetric spaces in Lie groups

In this paper we give a realization of some symmetric space G/K as a closed submanifold P of G. We also give several equivalent representations of the submanifold P. Some properties of the set gK∩P are also discussed, where gK is a coset space in G.     

On Anosov automorphisms of nilmanifolds

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that n⊕?⊕n (s times, s≥2) has an Anosov rational form for any graded real nilpotent Lie algebra n having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types (5,3) and (3,3,2) are not possible for Anosov Lie algebras.     

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.     

Modifying a branched surface to carry a foliation

We consider C¹ nonsingular flows on a closed 3-manifold under which there is no transverse disk that flows continuously back into its own interior. We provide an algorithm for modifying any branched surface transverse to such a flow ? that terminates in a branched surface carrying a foliation F precisely when F is transverse to ?. As a corollary, we find branched surfaces that do not carry foliations but that lift to ones that do.     

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.