A simple proof of polar decomposition in pseudo-Euclidean geometry
We give a simple direct proof of the polar decomposition for separated linear maps in pseudo-Euclidean geometry.
Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows
We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.
Abundance of hyperbolicity in the topology
Asymptotic windings over the trefoil knot.
Consider the group G:=PSL2(R) and its subgroups Γ:= PSL2(Z) and Γ':=DSL2(Z). G/Γ is a canonical realization (up to an homeomorphism) of the complement S3T of the trefoil knot T, and G/Γ' is a canonical realization of the 6-fold branched cyclic cover of S3T, which has a 3-dimensional cohomology of 1-forms.Putting natural left-invariant Riemannian metrics on G, it makes sense to ask which is the asymptotic homology performed by the Brownian motion in G/Γ', describing thereby in an intrinsic way part...
Borel summation and splitting of separatrices for the Hénon map
We study two complex invariant manifolds associated with the parabolic fixed point of the area-preserving Hénon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the “first” singularity and prove that a constant, which describes the splitting of the invariant manifolds, does not vanish. An interpretation...
Complex one-frequency cocycles
We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other...
Density of hyperbolicity and tangencies in sectional dissipative regions
Entropy-expansiveness and domination for surface diffeomorphisms.
Generic linear cocycles over a minimal base
We prove that a generic linear cocycle over a minimal base dynamics of finite dimension has the property that the Oseledets splitting with respect to any invariant probability coincides almost everywhere with the finest dominated splitting. Therefore the restriction of the generic cocycle to a subbundle of the finest dominated splitting is uniformly subexponentially quasiconformal. This extends a previous result for SL(2,ℝ)-cocycles due to Avila and the author.
Geometric expansion, Lyapunov exponents and foliations
Heterodimensional cycles, partial hyperbolicity and limit dynamics
We study one-parameter families of diffeomorphisms unfolding heterodimensional cycles (i.e. cycles containing periodic points of different indices). We construct an open set of such arcs such that, for a subset of the parameter space with positive relative density at the bifurcation value, the resulting nonwandering set is the disjoint union of two hyperbolic basic sets of different indices and a strong partially hyperbolic set which is robustly transitive. The dynamics of the diffeomorphisms we...
Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows
Multidimensional singular -lemma.
Nonuniform center bunching and the genericity of ergodicity among partially hyperbolic symplectomorphisms
We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns–Wilkinson and Avila–Santamaria–Viana. Combining this new technique with other constructions we prove that -generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.
On maximal transitive sets of generic diffeomorphisms
Partial hyperbolicity and homoclinic tangencies
We show that any diffeomorphism of a compact manifold can be approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.
Partial hyperbolicity for symplectic diffeomorphisms
Poincaré-Hopf index and partial hyperbolicity
We use the theory of partially hyperbolic systems [HPS] in order to find singularities of index for vector fields with isolated zeroes in a -ball. Indeed, we prove that such zeroes exists provided the maximal invariant set in the ball is partially hyperbolic, with volume expanding central subbundle, and the strong stable manifolds of the singularities are unknotted in the ball.
Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions
We study a partially hyperbolic and topologically transitive local diffeomorphism F that is a skew-product over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The associated transitive invariant set Λ possesses a very rich fiber structure, it contains uncountably many trivial and uncountably many non-trivial fibers. Moreover, the spectrum of the central Lyapunov exponents of...
Regular projectively Anosov flows on three-dimensional manifolds
We give the complete classification of regular projectively Anosov flows on closed three-dimensional manifolds. More precisely, we show that such a flow must be either an Anosov flow or decomposed into a finite union of -models. We also apply our method to rigidity problems of some group actions.