Related papers: Hyperbolic systems with non-diagonalisable princip…
We prove that the statistical properties of random perturbations of a nonuniformly hyperbolic diffeomorphism are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic…
A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a…
In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full…
We present a local almost everywhere regularity result for a general nonlinear non-diagonal parabolic system, which main part depends on symmetric part of the gradient.
We prove that every $C^2$ conservative partially hyperbolic diffeomorphism of a closed 3-manifold without periodic points is ergodic, which gives an affirmative answer to the Ergodicity Conjecture by Hertz-Hertz-Ures in the absence of…
In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this…
This paper is devoted to discuss the stabilizability of a class of $ 2 \times2 $ non-homogeneous hyperbolic systems. Motivated by the example in \cite[Page 197]{CB2016}, we analyze the influence of the interval length $L$ on stabilizability…
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric…
A {\em singular hyperbolic set} is a partially hyperbolic set with singularities (all hyperbolic) and volume expanding central direction \cite{MPP1}. We study connected, singular-hyperbolic, attracting sets with dense closed orbits {\em and…
We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ in the flat unit torus $\Bbb T^\nu$, $\nu\ge2$. In the case $\nu=2$ we prove (the full hyperbolicity and) the ergodicity of such…
We provide two robust examples of globally partially hyperbolic systems with a multi one-dimensional center splitting, for which all Gibbs u-states are hyperbolic and the number of physical measures is fixed. In the second example, the…
We explicitly construct a dynamically incoherent partially hyperbolic endomorphisms of $\mathbb{T}^2$ in the homotopy class of any linear expanding map with integer eigenvalues. These examples exhibit branching of centre curves along…
We show that a $\mathrm{GL}(d,\mathbb{R})$ cocycle over a hyperbolic system with constant periodic data has a dominated splitting whenever the periodic data indicates it should. This implies global periodic data rigidity of generic Anosov…
We are concerned with viscous profiles (travelling waves and steady solutions) for mixed hyperbolic-parabolic systems in one space variable. For a class of systems including the compressible Navier Stokes equation, these profiles satisfy a…
We present the general properties of dynamic dissipative fluid distribution endowed with hyperbolical symmetry. All the equations required for its analysis are exhibited and used to contrast the behavior of the system with the spherically…
We consider the system of N (\ge 2) elastically colliding hard balls with masses m_1,..., m_N, radius r, moving uniformly in the flat torus T_L^{\nu}= R^\nu/L \cdot Z^\nu, \nu \ge 2. It is proved here that the relevant Lyapunov exponents of…
For 2-d hyperbolic systems with singularities, statistical properties are rather difficult to establish because of the fragmentation of the phase space by singular curves. In this paper, we construct a Markov partition of the phase space…
We consider dynamical systems generated by partially hyperbolic surface endomorphisms of class C^r with one-dimensional strongly unstable subbundle. As the main result, we prove that such a dynamical system generically admits finitely many…
We give examples of quasi-hyperbolic dynamical systems with the following properties : polynomial decay of correlations, convergence in law toward a non gaussian law of the ergodic sums (divided by $n^{3/4}$) associated to non degenerated…
We prove the well-posedness and regularity of solutions in mixed-norm weighted Sobolev spaces for a class of second-order parabolic and elliptic systems in divergence form in the half-space $\mathbb{R}^d_+ = \{x_d > 0\}$ subject to the…