A Riemannian metric on polynomial hyperbolic components

Critical exponent and Hausdorff dimension in pseudo-Riemannian hyperbolic geometry

The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\mathrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud and Kassel, called $\mathbb{H}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in $\mathbb{H}^{2,1}=\mathrm{ADS}^3$ which can be understood as a Lorentzian version of a famous Theorem of…

Non-hyperbolic behavior of geodesic flows of rank 1 surfaces

We prove that for the geodesic flow of a rank 1 Riemannian surface which is expansive but not Anosov the Hausdorff dimension of the set of vectors with only zero Lyapunov exponents is large.

Stably weakly shadowing symplectomorphisms are partially hyperbolic

Let M be a closed, symplectic connected Riemannian manifold, f a symplectomorphism on M. We prove that if f is C1-stably weakly shadowing on M, then the whole manifold M admits a partially hyperbolic splitting.

No hyperbolic sets in J_\infty for infinitely renormalizable quadratic polynomials

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.

平面多项式向量场中非双曲极限环的结构不稳定性研究

On the structural instability of non-hyperbolic limit cycles on planar polynomial vector fields

该研究聚焦平面多项式向量场非双曲极限环的结构稳定性问题，证明在惠特尼拓扑下奇数次非双曲极限环也存在结构不稳定性，补全了多项式情形下的相关结论。

Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing

We consider some nonuniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs-Markov-Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball B(x, r) converges to a Poisson distribution as the radius r $\to$ 0 and after suitable normalization.

Hyperbolic SRB measures and the leaf condition

Let $M$ be a Riemannian, boundaryless, and compact manifold, with $\dim M\geq 2$ and let $f$ be a $C^{1+}$ diffeomorphism. We show that there is a hyperbolic SRB measure if and only if there exists an unstable leaf with a subset of positive leaf volume of hyperbolic points which return to some Pesin set with positive frequency. This answers a question of Pesin.

Coexistence of critical orbit types in sub-hyperbolic polynomial maps

We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex numbers {\bf C} to itself which have degree two or more in each copy. As a consequence of these results we prove a transitivity relation between hyperbolic components in parameter space which was conjectured by Milnor.

Flat traces for a random partially hyperbolic map

We consider a $\mathbb R/\mathbb Z$ extension of an Anosov diffemorphism of a compact Riemannian manifold by a random function $τ$ and show that the flat traces of the transfer operator, reduced with respect to frequency in the fibers, converge in law towards Gaussians, up to an Ehrenfest time that decreases with the regularity of $τ$.

Topological structure of (partially) hyperbolic sets with positive volume

We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose differentiability is bigger than one. We show in particular that there are no partially hyperbolic horseshoes with positive volume for such diffeomorphisms. We also give a description of the limit set of almost every point belonging to a hyperbolic set or a partially hyperbolic set with positive volume.

一种动态算法用于多项式时间内计算双曲Julia集

A dynamical algorithm to compute hyperbolic Julia sets in polynomial time

本文提出一种动态算法，用于在多项式时间内计算复多项式的双曲Julia集，并确立其下可计算性。

Variation of the Liouville measure of a hyperbolic surface

For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Holder distributions for the geodesic…

Hyperbolic components and cubic polynomials

In the space of cubic polynomials, Milnor defined a notable curve $\mathcal S_p$, consisting of cubic polynomials with a periodic critical point, whose period is exactly $p$. In this paper, we show that for any integer $p\geq 1$, any bounded hyperbolic component on $\mathcal{S}_p$ is a Jordan disk.

三次多项式在超几何成分边界上的实层叠

Real Laminations of Cubic Polynomials on Boundaries of Hyperbolic Components

本文研究了三次多项式在超几何成分边界上的实层叠特性，并证明除了类型D外，所有超几何三次多项式均不具备组合刚性。