Equilibrium states for certain partially hyperbolic attractors

Ergodic Theory and Dynamical Systems, 2004

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 singular hyperbolic attractor. In this paper we study the perturbations of singular hyperbolic attractors for three-dimensional flows. It is proved that any attractor obtained from such perturbations contains

2013

We analyze a phase-field system where the energy balance equation is linearly coupled with a nonlinear and nonlocal ODE for the order parameter χ. The latter equation is characterized by a space convolution term which models particle interaction and a singular configuration potential that forces χ to take values in (−1, 1). We prove that the corresponding dynamical system has a bounded absorbing set in a suitable phase space. Then we establish the existence of a finite-dimensional global attractor.

Pacific Journal of Mathematics, 2004

A recent problem in dynamics is to determinate whether an attractor Λ of a C r flow X is C r robust transitive or not. By attractor we mean a transitive set to which all positive orbits close to it converge. An attractor is C r robust transitive (or C r robust for short) if it exhibits a neighborhood U such that the set ∩ t>0 Y t (U) is transitive for every flow Y C r close to X. We give sufficient conditions for robustness of attractors based on the following definitions. An attractor is singular-hyperbolic if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction [MPP]. An attractor is C r critically-robust if it exhibits a neighborhood U such that ∩ t>0 Y t (U) is in the closure of the closed orbits is every flow Y C r close to X. We show that on compact 3-manifolds all C r critically-robust singular-hyperbolic attractors with only one singularity are C r robust.

Communications in Mathematical Physics, 2000

We study a bifurcation of Axiom A (hyperbolic) vector fields in dimension three leading to robust strange attractors with singularities. The Axiom A vector fields involved in the bifurcation exhibit a basic set equivalent to the suspension of a three symbol subshift. The attractors arising from this kind of bifurcation are not equivalent to the geometric Lorenz attractors.

Transactions of the American Mathematical Society, 2012

On every compact 3-manifold, we build a non-empty open set U of Diff 1 (M ) such that, for every r ≥ 1, every C r -generic diffeomorphism f ∈ U ∩ Diff r (M ) has no topological attractors. On higher dimensional manifolds, one may require that f has neither topological attractors nor topological repellers. Our examples have finitely many quasi attractors. For flows, we may require that these quasi attractors contain singular points. Finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics. * This work has been done during the stays of Li Ming and Yang Dawei at the IMB, Université de Bourgogne and we thank the IMB for its warm hospitality. M. Li is supported by a post doctoral grant of the Région Bourgogne, and D. Yang is supported by CSC of Chinese Education Ministry.

Ergodic Theory and Dynamical Systems, 2010

We study ergodic properties of invariant measures for the partially hyperbolic horseshoes, introduced in Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys. 29 (2009), 433–474]. These maps have a one-dimensional center direction Ec, and are at the boundary of the (uniformly) hyperbolic diffeomorphisms (they are constructed bifurcating hyperbolic horseshoes via heterodimensional cycles). We prove that every ergodic measure is hyperbolic, but the set of Lyapunov exponents in the central direction has gap: all ergodic invariant measures have negative exponent, with the exception of one ergodic measure with positive exponent. As a consequence, we obtain the existence of equilibrium states for any continuous potential. We also prove that there exists a phase transition for the smooth family of potentials given by ϕt=t log ∣DF∣Ec∣.

Communications in Mathematical Physics, 2000

We study a bifurcation of Axiom A (hyperbolic) vector fields in dimension three leading to robust strange attractors with singularities. The Axiom A vector fields involved in the bifurcation exhibit a basic set equivalent to the suspension of a three symbol subshift. The attractors arising from this kind of bifurcation are not equivalent to the geometric Lorenz attractors.

2005

We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their orbits coincide. Secondly, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a $u$-Gibbs state and an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction. This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.

Proceedings of the American Mathematical Society

We state in a short way a result that improves one of the main theorems in a paper of M. Gobbino concerning the topological properties that the phase space induces in an attractor of a discrete dynamical system.