Loughborough University is a top-ten rated university in England for research intensity (REF2014) and an outstanding 66% of the work of Loughborough’s academic staff who were eligible to be submitted to the REF was judged as ‘world-leading’ or ‘internationally excellent’, compared to a national average figure of 43%. In choosing Loughborough for your research, you’ll work alongside academics who are leaders in their field. You will benefit from comprehensive support and guidance from our Doctoral College, including tailored careers advice, to help you succeed in your research and future career. The research will be conducted in the Department of Mathematical Sciences which works to deliver innovative research that has the potential to have significant impact on industry. Nonlinear dynamics is one of the main research themes in the Department. This includes integrable systems, dynamical systems, ergodic theory, stochastic systems, nonlinear PDEs, biological systems, waves and vortices in geophysical fluid dynamics. Research students within the department enjoy a vibrant learning community benefitting from a range of resources to support their work, plus the opportunity to undertake training in lecturing/tutoring. The study of rare events in dynamical systems is currently one of the most active branches of research in ergodic theory. In part, this is due to their interesting applications in earth and ocean sciences. An example of a rare event in dynamical systems appears in the study of ‘open’ systems where orbits infrequently escape from the domain, typically by falling into a small “hole” in the phase space. The goal of the project is to use spectral techniques (via the so called Perron-Frobenius operator) to approximate escape rates in open systems and to compute the Hausdorff dimension of survival sets.