Geometry and dynamics of systems with a hyperbolic flavor

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The goal of this proposal is to promote research and the training of HQPs in domains of theoretical mathematics at the intersection of dynamical systems, geometry and low dimensional topology. My research program focuses on two distinct subfamilies of questions. The first involves the understanding and topological classification of certain types of dynamical systems in manifolds of low dimension. The second is concerned with a range of questions regarding the interactions between the dynamical properties of systems of geometrical origin (the geodesic flow of a Finsler or Riemannian manifold being the foremost), and geometric or spectral data coming from the metric.******Both of these research domains have a long and fruitful history and my proposed research will answer some naturally arising new questions as well as revisit older open problems with new tools and ideas. This research will lead to a better understanding of the relationships between some dynamical systems and the geometrical or topological aspects of the ambient space. The tools used in my work are also applicable in many different domains. Both Finsler geometry and dynamical systems with some hyperbolic behaviour can be found, and used, for instance, in Physics, Biology, or Numerical Analysis.******A very general question, which has been studied in various forms for many years, is the following: Given a specific type of dynamical system, what properties must a manifold possess in order to support such a system? And, given one such manifold, we want to obtain a classification of all the "different" dynamical systems of the original type that exists on the manifold. In this proposal, I aim to make progress on that question for two related types of dynamical systems -- Anosov flows and partially hyperbolic diffeomorphisms -- on 3- or 5-manifolds, as, for now at least, the low dimensional case is the only one where we can hope to obtain significant results. The existence of such dynamical systems, which are two stereotypical examples of systems with chaotic behaviour, have already been shown to be intrinsically linked to some fine topological properties of the manifolds supporting them. My research will lead to a deeper understanding of these relationships.******The second topic highlighted in this proposal is concerned with Finsler or Riemannian metrics with hyperbolic behaviour (defined here as either negatively curved metrics, or more generally, metrics with Anosov geodesic flow) and revolves around the generic question of how "similar" two metrics have to be to have the same unmarked length spectrum (i.e., the length of their closed geodesics, counted with multiplicity, are the same). This type of question (think of the famous "Can one hear the shape of a drum?") has previously led to some significant achievements and my work will further our knowledge of the relationships between some dynamical and geometrical data.