Sharp polynomial bounds on the number of Pollicott–Ruelle resonances

We give a sharp polynomial bound on the number of Pollicott–Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure and Sjöstrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer and Sjöstrand in scattering theory and by Faure and Sjöstrand in the theory of Anosov flows.

On the Differential Geometry of Flows in Nonlinear Dynamical Systems

In order to investigate the geometrical relation between two flows in two dynamical systems, a flow for an investigated dynamical system is called the compared flow and a flow for a given dynamical system is called the reference flow. A surface on which the reference flow lies is termed the reference surface. The time-change rate of the normal distance between the reference and compared flows in the normal direction of the reference surface is measured by a new function (i.e., G function). Based on the surface of the reference flow, the kth-order G functions are introduced for the noncontact and lth-order contact flows in two different dynamical systems. Through the new functions, the geometric relations between two flows in two dynamical systems are investigated without contact between the reference and compared flows. The dynamics for the compared flow with a contact to the reference surface is briefly addressed. Finally, the brief discussion of applications is given.

On the Differential Geometry of Flows in Nonlinear Dynamical Systems

In this paper, in order to investigate the relation between two flows given in two dynamical systems, a flow for an investigated dynamical system is called the compared flow and a flow for a given dynamical system is called the reference flow. A surface on which the reference flow lies is termed the reference surface. For a small time interval, the change rate of the normal distance between the reference and compared flows on the normal direction of the reference surface is measured by a new function (i.e., G-function). Based on the surface of the reference flow, the kth -order G-functions for the non-contact and lth-order contact flows in two different dynamical systems are introduced. Through the new functions, the geometric relations between two flows in two dynamical systems imposed in the same phase space are investigated without contact between the reference and compared flows. The compared flow passing through, returning back from and paralleling to the surface of the reference flow is discussed first. The tangency and passability for a compared flow to the surface of a reference flow with the lth-order contact are presented. The dynamics for the compared flow with such a contact to the reference surface is briefly addressed. Finally, the brief discussion of applications is given. The contact and tangential singularity between two flows are different. The kth-order contact of the compared flow to the reference surface indicates that the compared flow to the reference surface is of the kth-order singularity. However, the compared flow with the kth-order singularity to the reference surface does not mean that the compared flow to the reference surface is of the kth-order contact.

Convex Hamiltonians without conjugate points

We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.