Related papers: Geometric Expansions, Lyapunov Exponents and Folia…

We study $3$-dimensional partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the geometry and dynamics of Burago and Ivanov's center stable and center unstable \emph{branching} foliations. This extends our…

We prove that for any partially hyperbolic diffeomorphism with one dimensional neutral center on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable…

For partially hyperbolic diffeomorphisms with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton{a set consisting of finitely many hyperbolic periodic points with maximal cardinality for…

We study harmonic and totally invariant measures in a foliated compact Riemannian manifold isometrically embedded in an Euclidean space. We introduce geometrical techniques for stochastic calculus in this space. In particular, using these…

Let $f$ be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu$. We relate those entropies to covering numbers in order to give a new upper bound on…

We construct examples illustrating that dynamically-defined distributions of holomorphic diffeomorphisms on compact complex manifolds are not necessarily holomorphic in any open subset. More precisely, for any $n\geq 5$, we construct a…

We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states…

We give sufficient conditions for an expansive partially hyperbolic diffeomorphism with one-dimensional center to be (topologically) Anosov.

In this paper, we introduce the unstable topological pressure for C^1-smooth partially hyperbolic diffeomorphisms with sub-additive potentials. Moreover, without any additional assumption, we have established the expected variational…

We prove that the Lyapunov exponents, cosidered as functions of measures with non compact support, are semicontinuous with respect to the Wasserstein topology but not with respect to the weak* topology. Moreover, we prove that they are not…

We show that if an endomorphism $f:\mathbb{T}^2 \to \mathbb{T}^2$ is absolutely partially hyperbolic, then it has a center foliation. Moreover, the center foliation is leaf conjugate to that of its linearization.

We introduce a class of orbits which may have $0$ Lyapunov exponents, but still demonstrate some sensitivity to initial conditions. We construct a countable Markov partition with a finite-to-one almost everywhere induced coding, and which…

Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behaviour of almost every…

In the case of smooth non-invertible maps which are hyperbolic on folded basic sets $\Lambda$, we give approximations for the Gibbs states (equilibrium measures) of arbitrary H\"{o}lder potentials, with the help of weighted sums of atomic…

In this work, we deal with a notion of partially hyperbolic endomorphism. We explore topological properties of this definition and we obtain, among other results, obstructions to get center leaf conjugacy with the linear part, for a class…

We consider quasilinear parabolic evolution equations in the situation where the set of equilibria forms a finite-dimensional C^1-manifold which is normally hyperbolic. The existence of foliations of the stable and unstable manifolds is…

We study the existence and regularity of invariant graphs for bundle maps (or bundle correspondences with generating bundle maps motivated by ill-posed differential equations) having some relative partial hyperbolicity on non-trivial and…

Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are H\"older continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that…

The statistical properties of mostly expanding partially hyperbolic diffeomorphisms have been substantially studied. In this paper, we would like to address the entropy properties of mostly expanding partially hyperbolic diffeomorphisms. We…

We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a…