Strong Unstable Foliation Research Articles

Horospherical dynamics in invariant subvarieties

We consider the horospherical foliation on any invariant subvariety in the moduli space of translation surfaces. This foliation can be described dynamically as the strong unstable foliation for the geodesic flow on the invariant subvariety, and geometrically, it is induced by the canonical splitting of C-valued cohomology into its real and imaginary parts. We define a natural volume form on the leaves of this foliation, and define horospherical measures as those measures whose conditional measures on leaves are given by the volume form. We show that the natural measures on invariant subvarieties, and in particular, the Masur-Veech measures on strata, are horospherical. We show that these measures are the unique horospherical measures giving zero mass to the set of surfaces with horizontal saddle connections, extending work of Lindenstrauss-Mirzakhani and Hamenstädt for principal strata. We describe all the leaf closures for the horospherical foliation.

Robust minimality of strong foliations for DA diffeomorphisms: 𝑐𝑢-volume expansion and new examples

Let f f be a C 2 C^2 partially hyperbolic diffeomorphisms of T 3 \mathbb {T}^3 (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism A A with eigenvalues λ s > 1 > λ c > λ u . \begin{equation*} \lambda _{s}>1>\lambda _{c}>\lambda _{u}. \end{equation*} Under the assumption that the set { x : ∣ log ⁡ det ( T f ∣ E c u ( x ) ) ∣≤ log ⁡ λ u } \begin{equation*} \{x: \,\mid \log \det (Tf\mid _{E^{cu}(x)})\mid \leq \log \lambda _{u} \} \end{equation*} has zero volume inside any unstable leaf of f f where E c u = E c ⊕ E u E^{cu} = E^c\oplus E^u is the center unstable bundle, we prove that the stable foliation of f f is C 1 C^1 robustly minimal, i.e., the stable foliation of any diffeomorphism C 1 C^1 sufficiently close to f f is minimal. In particular, f f is robustly transitive. We build, with this criterion, a new example of a C 1 C^1 open set of partially hyperbolic diffeomorphisms, for which the strong stable foliation and the strong unstable foliation are both minimal.

A note on rigidity of Anosov diffeomorphisms of the three torus

We consider Anosov diffeomorphisms on $\mathbb{T}^3$ such that the tangent bundle splits into three subbundles $E^s_f \oplus E^{wu}_f \oplus E^{su}_f.$ We show that if $f$ is $C^r, r \geq 2,$ volume preserving, then $f$ is $C^1$ conjugated with its linear part $A$ if and only if the center foliation $\mathcal{F}^{wu}_f$ is absolutely continuous and the equality $\lambda^{wu}_f(x) = \lambda^{wu}_A,$ between center Lyapunov exponents of $f$ and $A,$ holds for $m$ a.e. $x \in \mathbb{T}^3.$ We also conclude rigidity of derived from Anosov diffeomorphism, assuming an strong absolute continuity property (Uniform bounded density property) of strong stable and strong unstable foliations.

Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse

We study billiard dynamics inside an ellipse for which the axes lengths are changed periodically in time and an -small quartic polynomial deformation is added to the boundary. In this situation the energy of the particle in the billiard is no longer conserved. We show a Fermi acceleration in such system: there exists a billiard trajectory on which the energy tends to infinity. The construction is based on the analysis of dynamics in the phase space near a homoclinic intersection of the stable and unstable manifolds of the normally hyperbolic invariant cylinder Λ, parameterised by the energy and time, that corresponds to the motion along the major axis of the ellipse. The proof depends on the reduction of the billiard map near the homoclinic channel to an iterated function system comprised by the shifts along two Hamiltonian flows defined on Λ. The two flows approximate the so-called inner and scattering maps, which are basic tools that arise in the studies of the Arnold diffusion; the scattering maps defined by the projection along the strong stable and strong unstable foliations Wss,uu of the stable and unstable invariant manifolds at the homoclinic points. Melnikov type calculations imply that the behaviour of the scattering map in this problem is quite unusual: it is only defined on a small subset of Λ that shrinks, in the large energy limit, to a set of parallel lines t = const as .

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.

Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation

Given a finite volume negatively curved Riemannian manifold M , we give a precise relation between the logarithmic growth rates of the excursions of the strong unstable leaves of negatively recurrent unit tangent vectors into cusp neighborhoods of M and their linear divergence rates under the geodesic flow. Our results hold in the more general setting where M is the quotient of any proper CAT(±1) metric space X by any geometrically finite discrete group of isometries of X . As an application to non-Archimedian Diophantine approximation in positive characteristic, we relate the growth of the orbits of \mathcal{O}_{\hat K} -lattices under one-parameter unipotent subgroups of \mathrm{GL}_2(\hat K) with approximation exponents and continued fraction expansions of elements of the local field \hat K of formal Laurent series over a finite field.

Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances

This paper deals with interactions between metric quasiconformal geometry and the rigidity of Anosov flows. In the first part of this article, we study a canonical time change of Anosov flows. Then we use it to obtain the thorough classification of volume-preserving quasiconformal Anosov flows. Given a $C^\infty$ Anosov flow $\varphi$, it is well-known that along the leaves of its strong unstable foliation there are two natural distances: the Hamenst$\ddot {\rm a}$dt distance and the induced Riemannian distance. In general these two distances are H$\ddot{\rm o}$lder equivalent. We prove the following rigidity result about quasisymmetric equivalence: Let $\varphi: M\to M$ be a $C^\infty$ transversely symplectic Anosov flow with $C^1$ weak distributions. We suppose that ${\rm dim}M\geq 5$. &nbsp If over a leaf of the strong unstable foliation, the Hamenst$\ddot{\rm a}$dt distance is quasisymmetric equivalent to the Riemannian distance, then up to finite covers $\varphi$ is $C^\infty$ orbit equivalent either to the geodesic flow of a closed hyperbolic manifold, or to the suspension of an Anosov automorphism.

MINIMALITY OF STRONG STABLE AND UNSTABLE FOLIATIONS FOR PARTIALLY HYPERBOLIC DIFFEOMORPHISMS

We give a topological criterion for the minimality of the strong unstable (or stable) foliation of robustly transitive partially hyperbolic diffeomorphisms.As a consequence we prove that, for $3$-manifolds, there is an open and dense subset of robustly transitive diffeomorphisms (far from homoclinic tangencies) such that either the strong stable or the strong unstable foliation is robustly minimal.We also give a topological condition (existence of a central periodic compact leaf) guaranteeing (for an open and dense subset) the simultaneous minimality of the two strong foliations.AMS 2000 Mathematics subject classification: Primary 37D25; 37C70; 37C20; 37C29

Box Dimensions and Topological Pressure for some Expanding Maps

We consider an expanding map defined on an open region of the plane and study the box dimensions of its invariant sets. Under the condition that the map leaves invariant a “strong unstable foliation”\(\), we prove that the box dimension of an invariant set is given by δF+δT, where δT is its dimension transverse to \(\) and δF is the root of a certain function involving topological pressure.

ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS

An analogue of the center manifold theory is proposed for non-local bifurcations of homo- and heteroclinic contours. In contrast with the local bifurcation theory it is shown that the dimension of non-local bifurcational problems is determined by the three different integers: the geometrical dimension dg which is equal to the dimension of a non-local analogue of the center manifold, the critical dimension dc which is equal to the difference between the dimension of phase space and the sum of dimensions of leaves of associated strong-stable and strong-unstable foliations, and the Lyapunov dimension dL which is equal to the maximal possible number of zero Lyapunov exponents for the orbits arising at the bifurcation. For a wide class of bifurcational problems (the so-called semi-local bifurcations) these three values are shown to be effectively computed. For the orbits arising at the bifurcations, effective restrictions for the maximal and minimal numbers of positive and negative Lyapunov exponents (correspondingly, for the maximal and minimal possible dimensions of the stable and unstable manifolds) are obtained, involving the values dc and dL. A connection with the problem of hyperchaos is discussed.

Spherical means on compact Riemannian manifolds of negative curvature

We show that the spherical mean of functions on the unit tangent bundle of a compact manifold of negative curvature converges to a measure containing a vast amount of information about the asymptotic geometry of those manifolds. This measure is related to the unique invariant measure for the strong unstable foliation, as well as the Patterson-Sullivan measure at infinity. It turns out to be invariant under the geodesic flow if and only if the mean curvature of the horospheres is constant. We use this measure in the study of rigidity problems.

A new description of the Bowen—Margulis measure

AbstractThe Bowen-Margulis measure on the unit tangent bundle of the universal covering of a compact manifold of negative curvature is determined by its restriction to the leaves of the strong unstable foliation. We describe this restriction to any strong unstable manifold W as a spherical measure with respect to a natural distance on W.

Margulis distributions for Anosov flows

For the strong unstable foliation (or horocycle foliation) of an Anosov flow there exists a unique transverse measure called the Margulis measure. In this paper we extend Margulis' results to more general “transverse distributions” for the foliation. As an application we derive our main result: The non-zero analytic extension to a strip of the Selberg zeta function for compact surfaces of constant negative curvature persists under small perturbations in the metric. There is an equivalent formulation in terms of the Fourier transform of the correlation function.

Anosov flows on new three manifolds

Throughout this paper, Anosov flows ~t:M ~M are volume preserving (a nontrivial assumption in light of Franks and Williams' recent example), and their ambient manifolds M are compact, orientable and three dimensional. Our purpose is to present a simple construction that yields a family of C ~ smooth, nonalgebraic (definitions below) Anosov flows. The existence of a volume-preserving Anosov flow ~ on a given manifold M 3 has immediate corollaries. The stable and unstable foliations are codimension one and minimal. By reparameterizing �9 if necessary, the flows (called horocycle flows) generated by the strong stable and strong unstable foliations are also minimal. We therefore have the relations

Unique ergodicity for horocycle foliations

The notion of unique ergodicity is extended to multidimensional foliations, and it is shown that ifg is the strong stable or strong unstable foliation of a topologically mixing basic set Ωκ for an Axiom A diffeomorphism or flow theng is uniquely ergodic.