This project aims to bring Dr Jonathan Meddaugh, one of the most promising young dynamicists of his generation, to the University of Birmingham to work with Dr Chris Good on a project that aims to capitalize on recent exciting developments in the theory of dynamical systems. Key to the analysis of dynamical systems are the asymptotic behaviour captured by limit sets and an understanding how computed (approximate) orbits correspond to true orbits, i.e. how pseudo-orbits are shadowed by real orbits. In recent work, Meddaugh has shown that shadowing is closely linked to characterizations of limit sets in terms of internally chain transitivity developed by Good and his collaborators. This project will develop the general theory behind these links and, using innovative applications of non-Hausdorff topologies and symbolic dynamics, make significant contributions to our understanding of shadowing and limit sets in a number of important families of dynamical system, such as shift spaces and polynomial maps on their Julia sets. During the project Meddaugh will develop the professional skills that will place him in an ideal position to secure a leading academic position.