The goal of this research proposal is to solve some fundamental open problems in nonlinear Perron-Frobenius theory. These problems concern order-preserving homogeneous maps on finite dimensional cones and have a distinct dynamical flavour. We shall ask, am ong others, when such maps have an invariant ray in the interior of the cone and investigate the long-term behaviour of orbits of these maps. To analyse these problems we intend to use methods from dynamics of nonexpansive maps and from hyperbolic geometry.
By solving these problems we would make a substantial contribution towards creating a complete nonlinear Perron-Frobenius theory. Such a theory is not only of theoretical interest, but also has applications in other areas of mathematics that are of current interest; in particular, in the analysis of diffusion processes on fractals and in optimal control and game theory. As both Dr Lemmens and Dr Sparrow have considerable experience in this field, it is to be expected that significant progress will be made .
The fellowship would allow Dr Lemmens to intensively work with Dr Sparrow at Warwick University for two years and to expand his research experience. It would give him the opportunity to become an active research member of one of the world's leading groups in dynamical systems. By taking post-graduate courses on dynamical systems theory and hyperbolic geometry, Dr Lemmens would enrich his expertise, which would later allow him to work on a variety of pure and applied topics in dynamics. In addition, Dr Lemmens would take some courses from the Centre of Academic Practice at Warwick University to further train his complementary academic skills. After the fellowship it is to be expected that Dr Lemmens has sufficiently matured as a mathematician to obtain a permanent lecturer position.
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