Related papers: Constructing center-stable tori
We study the structure of the stable norm of Finsler metrics on the 2-torus with a focus to points of irrational slope. By our results, the stable norm detects KAM-tori and hyperbolicity in the geodesic flow. Moreover, we study the stable…
In this work we obtain some metric and ergodic properties of $C^{1+}$ partially hyperbolic diffeomorphisms with one-dimensional topological neutral center, mainly regarding the behavior of its center foliation. Based on a trichotomy for the…
We prove that all ergodic automorphisms of the $N$-dimensional torus with two dimensional center are stably ergodic. This includes all ergodic automorphisms in dimension $N\leq 5$ or $N=7$. This generalizes a previous result of…
We show that a partially hyperbolic system can have at most a finite number of compact center-stable submanifolds. We also give sufficient conditions for these submanifolds to exist and consider the question of whether they can intersect…
We consider three-dimensional diffeomorphisms having simultaneously heterodimensional cycles and heterodimensional tangencies associated to saddle-foci. These cycles lead to a completely nondominated bifurcation setting. For every…
This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with…
Shub & Wilkinson and Ruelle & Wilkinson studied a class of volume preserving diffeomorphisms on the three dimensional torus that are stably ergodic. The diffeomorphisms are partially hyperbolic and admit an invariant central foliation of…
We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose…
We first present a general construction of Liouville domains as partial mapping tori. Then we study two examples where the (partial) monodromies exhibit certain hyperbolic behavior in the sense of Dynamical Systems. The first example is…
We introduce a notion of autonomous dynamical systems and apply it to prove rigidity of partially hyperbolic diffeomorphisms on closed compact three-manifolds under some smoothness hypothesis of their associated framing.
We show that every transitive dynamically coherent partially hyperbolic diffeomorphism with a one-dimensional center foliation $\W^c$ satisfying that $f(W)=W$ for every leaf $W\in \W^c$ is a discretized Anosov flow.
An earlier article with Francis Bonahon introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmuller space. We explicity compute these quantum hyperbolic invariants in…
We consider autonomous Hamiltonian systems and present an algorithm to compute at the same time partially hyperbolic invariant tori (whiskered tori), as well as high-order expansions of their stable and unstable manifolds. Such whiskered…
We discuss recent progress in understanding the dynamical properties of partially hyperbolic diffeomorphisms that preserve volume. The main topics addressed are density of stable ergodicity and stable accessibility, center Lyapunov…
We consider partially hyperbolic diffeomorphisms $f$ with a one-dimensional central direction such that the unstable entropy exceeds the stable entropy. Our main result proves that such maps have a finite number of ergodic measures of…
We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism $f$ on a closed $3$-manifold with exponential growth in its fundamental group such that $f^n$ is not isotopic to the identity for all $n\neq…
We develop an iterative technique for computing the unstable and stable eigenfunctions of the invariant tori of diffeomorphisms. Using the approach of Jorba, the linearized equations are rewritten as a generalized eigenvalue problem.…
We study $3$-dimensional partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the geometry and dynamics of Burago and Ivanov's center stable and center unstable \emph{branching} foliations. This extends our…
We consider the quintic nonlinear Schr{\"o}dinger on the circle. By applying a Birkhoff procedure and a KAM theorem, we exihibit a three dimension invariant torus that is linearly unstable. In comparison, we also prove that two dimensional…
We provide sufficient conditions on integrable analytic Hamiltonians that guarantee the existence, under arbitrary sufficiently small analytic perturbations, of invariant lower dimensional tori associated to an invariant resonant torus of…