### A note on pseudo-Anosov maps with small growth rate.

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We consider continuous $\mathrm{SL}(2,\mathbb{R})$-cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.

We study spectral properties of transfer operators for diffeomorphisms $T:X\to X$ on a Riemannian manifold $X$. Suppose that $\Omega $ is an isolated hyperbolic subset for $T$, with a compact isolating neighborhood $V\subset X$. We first introduce Banach spaces of distributions supported on $V$, which are anisotropic versions of the usual space of ${C}^{p}$ functions ${C}^{p}\left(V\right)$ and of the generalized Sobolev spaces ${W}^{p,t}\left(V\right)$, respectively. We then show that the transfer operators associated to $T$ and a smooth weight $g$ extend boundedly to these spaces, and...

We consider a family of planar systems introduced in 1991 by Benedicks and Carleson as a toy model for the dynamics of the so-called Hénon maps. We show that Smale’s Axiom A property is ${C}^{1}$-dense among the systems in this family, despite the existence of ${C}^{2}$-open subsets (closely related to the so-called Newhouse phenomena) where Smale’s Axiom A is violated. In particular, this provides some evidence towards Smale’s conjecture that Axiom A is a ${C}^{1}$-dense property among surface diffeomorphisms. The basic...

We show that the C¹-interior of the set of maps satisfying the following conditions: (i) periodic points are hyperbolic, (ii) singular points belonging to the nonwandering set are sinks, coincides with the set of Axiom A maps having the no cycle property.

The notion of C¹-stably positively expansive differentiable maps on closed ${C}^{\infty}$ manifolds is introduced, and it is proved that a differentiable map f is C¹-stably positively expansive if and only if f is expanding. Furthermore, for such maps, the ε-time dependent stability is shown. As a result, every expanding map is ε-time dependent stable.

La notion de type géométrique d’une partition de Markov est au centre de la classification des difféomorphismes de Smale i.e. des difféomorphismes ${C}^{1}$- structurellement stables des surfaces. On résout ici le problème de réalisabilité : on donne un critère effectif pour décider si une combinatoire abstraite est, ou n’est pas, le type géométrique d’une partition de Markov de pièce basique de difféomorphisme de Smale de surface compacte.

In this paper, we show that the C1 interior of the set of all continuum-wise expansive diffeomorphisms of a closed manifold coincides with the C1 interior of the set of all expansive diffeomorphisms. And the C1 interior of the set of all continuum-wise fully expansive diffeomorphisms on a surface is investigated. Furthermore, we have necessary and sufficient conditions for a diffeomorphism belonging to these open sets to be Anosov.

In this paper we prove a Central Limit Theorem for standard kernel estimates of the invariant density of one-dimensional dynamical systems. The two main steps of the proof of this theorem are the following: the study of rate of convergence for the variance of the estimator and a variation on the Lindeberg–Rio method. We also give an extension in the case of weakly dependent sequences in a sense introduced by Doukhan and Louhichi.

The weak shadowing property is really weaker than the shadowing property. It is proved that every element of the C¹ interior of the set of all diffeomorphisms on a ${C}^{\infty}$ closed surface having the weak shadowing property satisfies Axiom A and the no-cycle condition (this result does not generalize to higher dimensions), and that the non-wandering set of a diffeomorphism f belonging to the C¹ interior is finite if and only if f is Morse-Smale.