Non-accessible critical points of certain rational functions with Cremer points.
Non-existence of absolutely continuous invariant probabilities for exponential maps
We show that for entire maps of the form z ↦ λexp(z) such that the orbit of zero is bounded and Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.
Non-recurrent meromorphic functions
We consider a transcendental meromorphic function f belonging to the class ℬ (with bounded set of singular values). We show that if the Julia set J(f) is the whole complex plane ℂ, and the closure of the postcritical set P(f) is contained in B(0,R) ∪ {∞} and is disjoint from the set Crit(f) of critical points, then every compact and forward invariant set is hyperbolic, provided that it is disjoint from Crit(f). It is further shown, under general additional hypotheses, that f admits no measurable...
Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation
In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of , fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part which ensures that if such a perturbation of is formally conjugate to then it is also holomorphically conjugate to it. We study the normal form problem relatively to . We give a condition on that ensures that there...