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Register a new userIntransitive sectional-Anosov flows on 3-manifolds
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Publication date
2017
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English
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For each $ninmathbb{Z}^+$, we show the existence of Venice masks (i.e. intransitive sectional-Anosov flows with dense periodic orbits) containing $n$ equilibria on certain compact 3-manifolds. These examples are characterized because of the maximal invariant set is a finite union of homoclinic classes. Here, the intersection between two different homoclinic classes is contained in the closure of the union of unstable manifolds of the singularities.
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We prove that every sectional-Anosov flow of a compact 3-manifold $M$ exhibits a finite collection of hyperbolic attractors and singularities whose basins form a dense subset of $M$. Applications to the dynamics of sectional-Anosov flows on compact 3-manifolds include a characterization of essential hyperbolicity, sensitivity to the initial conditions (improving cite{ams}) and a relationship between the topology of the ambient manifold and the denseness of the basin of the singularities.
A {em sectional-Anosov flow} is a vector field on a compact manifold inwardly transverse to the boundary such that the maximal invariant set is sectional-hyperbolic (in the sense of cite{mm}). We prove that any $C^2$ transitive sectional-Anosov flow has a unique SRB measure which is stochastically stable under small random perturbations.
In this article, we give a quasi-final classification of quasiconformal Anosov flows. We deduce a very interesting differentable rigidity result for the orbit foliations of hyperbolic manifold of dimension at least three.
We classify quasiconformal Anosov flows whose strong stable and unstable distributions are at least two dimensional and the sum of these two distributions is smooth. We deduce from this classification result the complete classification of volume-preserving quasiconformal diffeomorphisms whose stable and unstable distributions are at least two dimensional. Our central idea is to take a good time change so that perodic orbits are equi-distributed with respect to a lebesgue measure.
We prove that for Anosov maps of the $3$-torus if the Lyapunov exponents of absolutely continuous measures in every direction are equal to the geometric growth of the invariant foliations then $f$ is $C^1$ conjugated to his linear part.
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