Related papers: Equilibrium states for certain partially hyperboli…

We prove the existence and uniqueness of tempered random attractors for stochastic Reaction-Diffusion equations on unbounded domains with multiplicative noise and deterministic non-autonomous forcing. We establish the periodicity of the…

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…

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the…

We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with…

We show that a partially hyperbolic system can have at most a finite number of compact center-stable submanifolds. We also give sufficient conditions for these submanifolds to exist and consider the question of whether they can intersect…

In this paper, we investigate the continuity of the attractors in time-dependent phase spaces. (i) We establish two abstract criteria on the upper semicontinuity and the residual continuity of the pullback $\mathscr D$-attractor with…

We study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting. Earlier work of Bonatti, Viana about existence and finitude of physical measures is extended to the case of local diffeomorphisms.…

We study small perturbations of a sectional hyperbolic set of a vector field on a compact manifold. Indeed, we obtain robustly finiteness of homoclinic classes on this scenary. Moreover, since attractor and repeller sets are particular…

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…

This paper gives a classification of partially hyperbolic systems in dimension 3 which have at least one torus tangent to the center-stable bundle.

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional,…

The goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations, as it was done for the semilinear case by Brunovsk\'y and Fiedler (1986), and generalized by Fiedler and Rocha (1996). In…

Equilibrium states for geodesic flows over closed rank 1 manifolds were studied recently by Burns, Climenhaga, Fisher and Thompson. For sufficiently regular potentials, it was shown that if the singular set does not carry full pressure then…

A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. The geometric Lorenz attractor \cite{gw} is an example of a…

We prove that at least one of the two invariant laminations of a strongly partially hyperbolic attractor with one-dimensional center bundle is minimal. This result extends those in [7] about minimal foliations for robustly transitive…

In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in \cite{DHRS07}. Despite the fact that the non-wandering set is a…

We analyze the long-time behavior of solutions to a Navier-Stokes-Cahn-Hilliard system with chemotaxis effects and a solution-dependent mass source term. The fluid velocity satisfies the Navier-Stokes system, the phase field variable…

There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, "hyperbolic". We show that for a sufficiently regular one-dimensional map satisfying a weak hyperbolicity assumption,…

For any integer $n \geq 5$, we construct an $n$-dimensional $C^1$ vector field exhibiting a robustly transitive singular attractor which is not sectional-hyperbolic. Nevertheless, the attractor is singular-hyperbolic. This provides the…

We prove the saturation of a generalized partially hyperbolic attractor of a $C^2$ map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a $C^1$-generic diffeomorphism has zero volume. In…