Related papers: Accessibility and centralizers for partially hyper…

We prove that a partially hyperbolic attracting set for a C2 vector field, having slow recurrence to equilibria, supports an ergodic physical/SRB measure if, and only if, the trapping region admits non-uniform sectional expansion on a…

We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which…

We study transitive partially hyperbolic diffeomorphisms in dimension 3 preserving a center foliation on which they act quasi-isometrically. We show that the diffeomorphism is up to finite lift and iterate, either a skew-product or a…

We study ergodic properties of compositions of holomorphic endomorphisms of the complex projective space chosen independently at random according to some probability distribution. Along the way, we construct positive closed currents which…

It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence hence for tameness. In this…

We prove the stochastic stability of an open class of partially hyperbolic diffeomorphisms, each of which admits two centers $E^c_1$ and $E^c_2$ such that any Gibbs $u$-state admits only positive (resp. negative) Lyapunov exponents along…

In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave…

For $C^{1+}$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an ergodic adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We then apply this to study…

Our recent work established existence and uniqueness results for $\mathcal{C}^{k,\alpha}_{\text{loc}}$ globally defined linearizing semiconjugacies for $\mathcal{C}^1$ flows having a globally attracting hyperbolic fixed point or periodic…

For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors. We…

We consider partially hyperbolic \( C^{1+} \) diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition \( E^s\oplus E^{cu} \). Assuming the existence of a set of…

We show that a $C^1-$generic non partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that…

A result on $C^0$ linearization which is differentiable at the hyperbolic fixed point is known. In this paper, we further investigate a partially hyperbolic diffeomorphism $F$ to find a local $C^0$ conjugacy, which is $C^1$ on the center…

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…

We prove that a partially hyperbolic attractor for a $C^1$ vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is $C^2$, and the center bundle admits the sectional expanding…

It was shown that in robustly transitive, partially hyperbolic diffeomorphisms on three dimensional closed manifolds, the strong stable or unstable foliation is minimal. In this article, we prove ``almost all'' leaves of both stable and…

We propose a criterion, referred to as order-n transversality, for transitivity of area preserving partially hyperbolic endomorphisms. Besides, we also give a further answer to the Gan's problem, as proposed in the work of Baolin He.

This article is devoted to the study of the historic set of ergodic averages in some nonuniformly hyperbolic systems. In particular, our results hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms…

We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak$*$-topology and in entropy. For hyperbolic ergodic measures, it is a classical…

We prove that for any $\ell\in\NN\cup\{\infty\}$ and any $r\in \NN$, every compact smooth Riemannian manifold $\cM$ of $\dim \cM\ge 5$ carries a $C^\infty$ volume preserving nonuniformly hyperbolic diffeomorphism, which has exactly $\ell$…