We study the microlocal kernel of h-pseudodifferential operators Oph(p) − z, where z belongs to some neighborhood of size O(h) of a critical value of its principal symbol p0(x, ξ). We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow Hp0 . First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in t...

We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbol...

We prove that if a diffeomorphism on a compact manifold preserves a nonatomic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measur...

employing a three critical points theorem, we prove the existence ofmultiple solutions for a class of neumann two-point boundary valuesturm-liouville type equations. using a local minimum theorem fordifferentiable functionals the existence of at least one non-trivialsolution is also ensured.

We consider dynamical systems on compact manifolds, which are local dif-feomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbo...

We prove that any C transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller [37] for maps of the interval with negative Schwarzian derivative. We also sh...