A complement to Hayashi Connecting Lemma

Correction to "Connecting invariant manifolds and the solution of the C^1 stability and Ω-stability conjectures for flows"

There is a gap in the proof of Lemma VII.4 in [Ann. of Math. (2) 145 (1997), 81--137]. We present an alternative proof of Theorem B (C^1 Omega-stable vector fields satisfy Axiom A). The novel and essential part in the proof of the stability and Omega-stability conjectures for flows is the connecting lemma introduced previously. A mistake in the proof of the last conjecture was pointed out to me by Toyoshiba, who later also provided an independent proof of it, again based on the connecting lemma and previous arguments by Mañé and Palis.

A connecting lemma for rational maps satisfying a no growth condition

We introduce and study a non uniform hyperbolicity condition for complex rational maps, that does not involve a growth condition. We call this condition Backward Contraction. We show this condition is weaker than the Collet-Eckmann condition, and than the summability condition with exponent~1. Our main result is a connecting lemma for Backward Contracting rational maps, roughly saying that we can perturb a rational map to connect each critical orbit in the Julia set with an orbit that does not accumulate on critical points. The proof of this result is based on Thurston's algorithm and some rig…

Sectional connecting lemma

A hyperbolic set on a compact manifold M, satisfies the property: given two of your any points p and q, such that for all positive ε>0, there is a trajectory in the hyperbolic set from a point ε-close to p to a point ε-close to q, then there is a point in M whose α-limit is that of p and whose ω-limit is that of q. Bautista and Morales give a version of this property, for sectional-Anosov flows (vector fields whose maximal invariant set is sectional-hyperbolic), including some conditions; among them that limit the dimension of M to three. In this paper, we prove a generalization of this result…

关于洛伦兹吸引子的C^r连接引理及其在ergodic测度空间中的应用

A $C^r$-connecting lemma for Lorenz attractors and its application on the space of ergodic measures

本文证明了关于洛伦兹吸引子的C^r连接引理，并展示了其在ergodic测度空间中的应用，揭示了不同条件下测度空间的连通性特性。

Perturbations of C1-diffeomorphisms and dynamics of generic conservative diffeomorphisms of surface

In the first part of this text we give a survey of the properties satisfied by the C1-generic conservative diffeomorphisms of compact surfaces. The main result that we will discuss is that a C1-generic conservative diffeomorphism of a connected compact surface is transitive. It is obtain as a consequence of a connecting lemma for pseudo-orbits. In the last parts we expose some recent developments of the C1-perturbation technics and the proof of this connecting lemma. We are not aimed to deal with technicalities nor to give the finest available versions of these results. Besides this theory exi…

$C^r$-链闭包引理对于某些部分双曲微分同胚

$C^r$-Chain closing lemma for certain partially hyperbolic diffeomorphisms

本文证明了部分双曲微分同胚的 $C^r$-链闭包引理，并由此得出通有情形下周期点稠密及链传递蕴含传递性。