In 1976 D. Sullivan gave an example of a flow on a compact manifold such that each one of its orbits is a circle and with the surprising property that there is no finite upper bound for their length. The aim of this article is to show that these type of examples do not appear as normally hyperbolic foliations. Namely, we prove that if a circle foliation is the center foliation of a (dynamically coherent) partially hyperbolic diffeormophism, then there is a finite upper bound for the length of the leaves. We also give short proofs of some dynamical consequences in the converse case: if the center foliation of a partially hyperbolic diffeomorphism $f$ is by compact leaves with uniformly bounded volume, then $f$ is dynamically coherent and plaque expansive.
We introduce some tools of symbolic dynamics to study the hyperbolic directions of partially hyperbolic diffeomorphisms, emulating the well known methods available for uniformly hyperbolic systems.
This paper provides a canonical compactification of the plane ${\mathbb R}^2$ by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather \cite{Ma}. Moreover any homeomorphism of ${\mathbb R}^2 $ preserving the foliations extends on the circle at infinity. Then this paper provides conditions ensuring the minimality of the action on the circle at infinity induced by an action on ${\mathbb R}^2 $ preserving one foliation or two transverse foliations. In particular the action on the circle at infinit…
We consider quasilinear parabolic evolution equations in the situation where the set of equilibria forms a finite-dimensional C^1-manifold which is normally hyperbolic. The existence of foliations of the stable and unstable manifolds is shown assuming merely C^1-regularity of the underlying equation.
We show that a smooth 1-parameter family of foliations by circles of a closed 3-manifold, deforming the foliation whose leaves are the fibers of a circle bundle, is trivial, i.e. all the foliations of the family arise from circle bundles isomorphic to the unperturbed one, if a continuity property of the Seifert leaves of the perturbation holds true. This rigidity property is true for any real analytic 1-parameter family of foliations by circles when the base space of the circle bundle defining the unperturbed foliation is not a torus. The dimensionality hypothesis is discussed in relation to a…
Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we give a new description of the universal circle of $\mathbb{R}$-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of $M$.
This paper investigates certain foliations of three-manifolds that are hybrids of fibrations over the circle with foliated circle bundles over surfaces: a 3-manifold slithers around the circle when its universal cover fibers over the circle so that deck transformations are bundle automorphisms. Examples include hyperbolic 3-manifolds of every possible homological type. We show that all such foliations admit transverse pseudo-Anosov flows, and that in the universal cover of the hyperbolic cases, the leaves limit to sphere-filling Peano curves. The skew R-covered Anosov foliations of Sergio Fenl…
Let (M, F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F . If every such function is constant on leaves we say that (M, F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.
In this paper we start a global study of the parameter space (dissipation, perturbation, frequency) of the dissipative spin-orbit problem in Celestial Mechanics with the aim of delimiting regions where the dynamics, or at least some of its important features, is determined. This is done via a study of the corresponding family of time $2π$-maps. In the same spirit as Chenciner in his 1895 article on bifurcations of elliptic fixed points, we are at first interested in delimiting regions where the normal hyperbolicity is sufficiently important to guarantee the persistence of an invariant attracti…
We study a class of scalar differential equations on the circle $S^1$. This class is characterized mainly by the property that any solution of such an equation possesses exponential dichotomy both on the semi-axes $\R_+$ and $\R_+$. Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations of this class are called gradient-like ones. As a result, we describe the global behavior of the foliation, introduce a complete invariant of uniform equivalency, give standard models for the equations of the distinguished class…
We consider smooth partially hyperbolic volume preserving Z^k actions on smooth manifolds, with uniformly compact center foliation. We show that under certain irreducibility condition on the action, bunching and uniform quasiconformality conditions, the action is a smooth fiber bundle extension of an Anosov action, or the center foliation is pathological. We obtain several corollaries of this result. For example, we prove a global dichotomy result that any smooth conservative circle extension over a maximal Cartan action is either essentially a product of an action by rotations and a linear An…
We numerically study quasiperiodic normally hyperbolic attracting invariant circles that appear for certain parameter values in a family of three-dimensional Henon-like maps. These parameter values make up contour segments in the parameter space where the invariant circles have constant irrational rotation number. At the edges of these segments we find a breakdown of the hyperbolicity of the invariant circle. We observe the collision and loss of smoothness of two of the invariant Lyapunov bundles while the Lyapunov exponents all remain distinct. This is very similar to the breakdown of quasipe…
We present and implement an algorithm for computing the invariant circle and the corresponding stable manifolds for 2-dimensional maps. The algorithm is based on the parameterization method, and it is backed up by an a-posteriori theorem established in [YdlL21]. The algorithm works irrespective of whether the internal dynamics in the invariant circle is a rotation or it is phase-locked. The algorithm converges quadratically and the number of operations and memory requirements for each step of the iteration is linear with respect to the size of the discretization. We also report on the result o…
A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered Anosov flow. The proof uses the universal circle to R-covered foliations.
We characterize the maximal entropy measures of partially hyperbolic C^2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons. In the special case of 3-dimensional nilmanifolds other than the torus, this entails the following dichotomy: either the diffeomorphism is a rotation extension of an Anosov diffeomorphism -- in which case there is a unique maximal measure, with full support and zero center Lyapunov exponents -- or there exist exactly two ergodic maximal measures, both hyperbolic and whose center Lyapunov …
We study asymptotically compact nonautonomous dynamical systems given by abstract cocycles in Banach spaces. Our main assumptions are given by a squeezing property in a quadratic cone field (given by a family of indefinite quadratic Lyapunov-like functionals) and asymptotic compactness. Under such conditions it is possible to reconstruct foliations as in the theory of normally hyperbolic manifolds. Our approach allows to unify many "practical" theories of local and nonlocal invariant manifolds, such as stable/unstable/center manifolds and inertial manifolds, and, moreover, it often leads to op…
We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We restrict our attention to continuous-time dynamical systems, or flows. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a $C^k$ disk bundle structure if the local stable foliation is assumed $C^k$. We the…
Let f:M->M be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of f. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves. Also we show that a finite cover of f fibers over an Anosov toral automorphism if one of the following conditions is met: 1. the center foliation of f has codimension 2, or 2. the center leaves of f are simply connected leaves and the unstable foliation of f is …
This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this flow assuming that the natural affine foliation is minimal too. We have tried to offer a simple presentation, which allows us to update and shed light on the classical theorem proved by G. A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this result to the product of PSL(2,R) and a Lie group G, whi…
For a class of partially hyperbolic $C^k$, $k>1$ diffeomorphisms with circle center leaves we prove existence and finiteness of physical (or Sinai-Ruelle-Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient manifold. Our conditions contain an open and dense subset of all $C^k$ partially hyperbolic skew-products on compact circle bundles. Our arguments blend ideas from the theory of Gibbs states for diffeomorphisms with mostly contracting center direction together with recent progress in the theory of cocycles over hyperbolic systems that call into play geometric pr…
We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on $\mathbb{A}^3$. Our approach can also be applied to more general Hamiltonians that are not necessarily convex. The main geometric objects in our framework are $3$-dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable m…
