Related papers: Removing zero Lyapunov exponents in volume-preserv…

We show that any topologically transitive codimension-one Anosov flow on a closed manifold is topologically equivalent to a smooth Anosov flow that preserves a smooth volume. By a classical theorem due to Verjovsky, any higher dimensional…

We give a new definition for a Lyapunov exponent (called new Lyapunov exponent) associated to a continuous map. Our first result states that these new exponents coincide with the usual Lyapunov exponents if the map is differentiable. Then,…

We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter…

We consider nonlinear hyperbolic conservation laws, posed on a differential (n+1)-manifold with boundary referred to as a spacetime, and in which the "flux" is defined as a flux field of n-forms depending on a parameter (the unknown…

We prove that for the geodesic flow of a rank 1 Riemannian surface which is expansive but not Anosov the Hausdorff dimension of the set of vectors with only zero Lyapunov exponents is large.

We prove that a C1-generic volume-preserving dynamical system (diffeomorphism or flow) has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, we prove that the C1-robustness,…

We prove three formulas for computing topological pressure of $C^1$-generic conservative diffeomorphism and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that…

We construct an example of a Hamiltonian flow $f^t$ on a $4$-dimensional smooth manifold $\mathcal{M}$ which after being restricted to an energy surface $\mathcal{M}_e$ demonstrates essential coexistence of regular and chaotic dynamics that…

For any $C^\infty$, area-preserving Anosov diffeomorphism $f$ of a surface, we show that a suspension flow over $f$ is $C^\infty$-conjugate to a constant-time suspension flow of a hyperbolic automorphism of the two torus if and only if the…

We study the normal forms for incompressible flows and maps in the neighborhood of an equilibrium or fixed point with a triple eigenvalue. We prove that when a divergence free vector field in $\mathbb{R}^3$ has nilpotent linearization with…

We consider a class of partially hyperbolic diffeomorphisms introduced in [BFP] which is open and closed and contains all known examples. If in addition the diffeomorphism is non-wandering, then we show it is accessible unless it contains a…

We show that the existence of physical measures for $C^\infty$ smooth instances of certain partially hyperbolic dynamics, both continuous and discrete, exhibiting mixed behavior (positive and negative Lyapunov exponents) along the central…

In this letter, we present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M, \omega). It is shown that every volume-preserving flow has some 2-forms acting the role of the Hamiltonian…

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…

We introduce a new measure of instability of area-preserving twist diffeomorphisms, which generalizes the notions of angle of splitting of separatrices, and flux through a gap of a Cantori. As an example of application, we establish a sharp…

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…

In this paper, we prove that for every $C^1$ vector field preserving an ergodic hyperbolic invariant measure which is not supported on singularities, if the Oseledec splitting of the ergodic hyperbolic invariant measure is a dominated…

In this paper, we prove that for $\mathcal{C}^1$ generic volume-preserving Anosov diffeomorphisms of a compact Riemannian manifold, Liv\v{s}ic measurable rigidity theorem holds. We also prove that for $\mathcal{C}^1$ generic…

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 introduce a finite volume scheme for the two-dimensional incompressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are respectively piecewise constant and affine. We use a projection…