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Register a new userOn the intersection of sectional-hyperbolic sets
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We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without such a hypothesis). Next we prove that, in general, such an intersection consists of a nonsingular hyperbolic set, finitely many singularities and regular orbits joining them. Afterward we exhibit a three-dimensional star flow with two homoclinic classes, one being positively (but not negatively) sectional-hyperbolic and the other negatively (but not positively) sectional-hyperbolic, whose intersection reduces to a single periodic orbit. This provides a counterexample to a conjecture by Shy, Zhu, Gan and Wen (cite{sgw}, cite{zgw}).
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We prove that every sectional-hyperbolic Lyapunov stable set contains a nontrivial homoclinic class.
Let f be an infinitely-renormalizable quadratic polynomial and J_infty be the intersection of forward orbits of small Julia sets of simple renormalizations of f. We prove that J_infty contains no hyperbolic sets.
Let $(X,mathscr{B}, mu,T,d)$ be a measure-preserving dynamical system with exponentially mixing property, and let $mu$ be an Ahlfors $s$-regular probability measure. The dynamical covering problem concerns the set $E(x)$ of points which are covered by the orbits of $xin X$ infinitely many times. We prove that the Hausdorff dimension of the intersection of $E(x)$ and any regular fractal $G$ equals $dim_{rm H}G+alpha-s$, where $alpha=dim_{rm H}E(x)$ $mu$--a.e. Moreover, we obtain the packing dimension of $E(x)cap G$ and an estimate for $dim_{rm H}(E(x)cap G)$ for any analytic set $G$.
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.
Let $mathcal{R}$ be a strongly compact $C^2$ map defined in an open subset of an infinite-dimensional Banach space such that the image of its derivative $D_F mathcal{R}$ is dense for every $F$. Let $Omega$ be a compact, forward invariant and partially hyperbolic set of $mathcal{R}$ such that $mathcal{R}colon Omegarightarrow Omega$ is onto. The $delta$-shadow $W^s_delta(Omega)$ of $Omega$ is the union of the sets $$W^s_delta(G)= {Fcolon dist(mathcal{R}^iF, mathcal{R}^iG) leq delta, for every igeq 0 },$$ where $G in Omega$. Suppose that $W^s_delta(Omega)$ has transversal empty interior, that is, for every $C^{1+Lip}$ $n$-dimensional manifold $M$ transversal to the distribution of dominated directions of $Omega$ and sufficiently close to $W^s_delta(Omega)$ we have that $Mcap W^s_delta(Omega)$ has empty interior in $M$. Here $n$ is the finite dimension of the strong unstable direction. We show that if $delta$ is small enough then $$cup_{igeq 0}mathcal{R}^{-i}W^s_{delta} (Omega)$$ intercepts a $C^k$-generic finite dimensional curve inside the Banach space in a set of parameters with zero Lebesgue measure, for every $kgeq 0$. This extends to infinite-dimensional dynamical systems previous studies on the Lebesgue measure of stable laminations of invariants sets.
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