Rigidity of magnetic and geodesic flows

In this thesis we will discus the flow of particles on a manifold, with and without the presence of a magnetic field. We will address two independent rigidity problems regarding this flows. The first problem relates to scattering boundary rigidity in the presence of a magnetic field. It has been shown in [DPSU] that, under some additional assumptions, two simple domains with the same scattering data are equivalent. We show that the simplicity of a region can be read from the metric in the boundary and the scattering data. This lets us extend the results in [DPSU] to regions with the same scattering data, where only one is known apriori to be simple. We will then use this results to resolve a local version of a question by Robert Bryant. That is, we show that a surface of constant curvature can not be modified in a small region while keeping all the curves of a fixed constant geodesic curvature closed. The second problem involves blocking properties of the geodesic flow. We say that a pair of points x and y is secure if there exists a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this part is to exhibit new examples of blocking phenomena both in the manifold and the billiard table setting. As an approach to this, we study if the product of secure configurations (or manifolds) is also secure. We introduce the concept of midpoint security that imposes that the geodesic reaches a blocking point exactly at its midpoint. We prove that products of midpoint secure configurations are midpoint secure. On the other hand, we give an example of a compact C1 surface that contains secure configurations that are not midpoint secure. This surface provides the first example of an insecure product of secure configurations, as well as billiard tables with similar blocking behavior