Related papers: Volume growth and entropy for $C^1$ partially hype…

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semi-conjugate to nonuniformly expanding maps. Using the theory of projective metric on cones we then prove…

We study generic diffeomorphisms with a homoclinc class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one…

For any $C^1$ diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the…

We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the…

In this work we study the class of mostly expanding partially hyperbolic diffeomorphisms. We prove that such class is $C^r$-open, $r>1$, among the partially hyperbolic diffeomorphisms (in the narrow sense) and we prove that the mostly…

We study the connection between the Lyapunov exponents and the volume growth of boundary distortion of regions in the phase space of the dynamical system.

We establish the relationship between the growth rate of periodic orbits and the topological entropy for $C^1$ generic vector fields: this extends a classical result of Katok for $C^{1+\alpha}(\alpha>0)$ surface diffeomorphisms to $C^1$…

We outline the flexibility program in smooth dynamics, focusing on flexibility of Lyapunov exponents for volume-preserving diffeomorphisms. We prove flexibility results for Anosov diffeomorphisms admitting dominated splittings into…

We show that any diffeomorphism of a compact manifold can be C1 approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.

We show that every codimension one partially hyperbolic diffeomorphism must support on $\mathbb{T}^{n}$. It is locally uniquely integrable and derived from a linear codimension one Anosov diffeomorphism. Moreover, this system is…

We give an upper bound for the growth of homology torsions of finite coverings of irreducible 3-manifolds with tori boundary in terms of hyperbolic volume.

We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove…

We use the Invariance Principle of Avila and Viana to prove that every partially hyperbolic symplectic diffeomorphism with 2-dimensional center bundle, and satisfying certain pinching and bunching conditions, can be $C^r$-approximated by…

For $C^1$ diffeomorphisms, we prove that the Pesin's entropy formula holds for some invariant measure supported on any topological attractor that admits a dominated splitting without mixed behavior. We also prove Shub's entropy conjecture…

We prove a generalization of a so called "invariance principle" for partially hyperbolic diffeomorphisms: if an invariant probability measure has all its center Lyapunov exponents equal to zero then the measure admits a center…

In this paper we establish a dichotomy for the ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with one-dimensional compact center leaves which are virtually skew products over (transitive) Anosov homeomorphism.…

We exhibit a local residual set of surface $C^1$ diffeomorphisms that are continuum-wise expansive but are not expansive. We also exhibit an open and dense set of surface $C^1$ diffeomorphisms where expansiveness implies being Anosov.

We consider $C^2$ Fr\'echet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite…

In this paper, we continue our investigation on sub-additive pressures for $C^1$-smooth partially hyperbolic diffeomorphisms. Under the assumption of unstable almost product property, we show that the unstable Bowen topological pressure on…

We construct measures of maximal $u$-entropy for any partially hyperbolic diffeomorphism that factors over an Anosov torus automorphism and has mostly contracting center direction. The space of such measures has a finite dimension, and its…