Dimension theory of dynamically defined sets

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My proposed programme of research lies at the intersection of dimension theory and dynamical systems, an exciting area which is concerned with the study of highly irregular geometric objects which arise naturally in the context of dynamical systems.Dynamical systems aims to give qualitative and quantitative information about the long term behaviour of collections of states which evolve over time according to some fixed rule. Often the collection of states which satisfy some prescribed statistical law form a fractal set; a geometric object which displays intricate detail at all magnifications. This link between the dynamical behaviour of a system and the fractal geometry of certain subsets has led to it being used to model natural phenomena such as cloud boundaries and fluid turbulence.The first objective in my programme of research is the study of self-similar sets with overlap, a research topic which lies at the heart of dimension theory, and has strong connections with ergodic theory, dynamical systems, and geometric measure theory. Such sets, despite being defined in a simple fashion, display intricate detail and often have zero area or volume, and so the tools of classical geometry prove to be not so useful. Fractal dimension quantifies how irregular an object is and provides an important tool with which to analyse these sets.Imposing a technical condition on these sets has enabled a rich theory to be developed that has shown deep relations with fields such as ergodic theory and dynamical systems. In the case that no such condition is imposed only partial results are known. It is my intention to investigate the case where no separation conditions are assumed with a view of focussing on necessary conditions for the coincidence of the Hausdorff and symbolic dimensions. Even partial results in this direction would have a profound effect on our understanding of self-similarity, and would spawn an entirely new line of research in this field. My second objective is to investigate the statistical properties of open dynamical systems. Often a dynamical system will display sensitive dependence on initial conditions, which means that the long term behaviour of an individual orbit is extremely difficult to predict. Motivated by a similar problem in statistical physics mathematicians developed the thermodynamic formalism, which provides an avenue to study the long term behaviour of typical trajectories in the dynamical system. My overarching goal within this field is to gain a better picture of the statistical properties of these this phenomena in the context of open dynamical systems. A novel application of this research would be to the field of computer science. Lempel-Ziv-Welsch is an algorithm used for compressing data, which operates by relating certain sub-blocks of data. The techniques involved with this objective would directly apply in this setting and would afford us a better picture of the efficiency of this algorithm. Such a result would be of great interest to those working in the field of computer science.The final objective for the programme is the study of diophantine approximation in a fractal setting. Diophantine approximation is concerned with how well one may approximate real numbers by rationals. The classical setting for this work has been in Euclidean space with the Lebesgue measure, with one of the many highlights being the celebrated theorem of Khinchin. Recently, there has been great interest in studying these problems for fractal sets and measures, with a view of proving an analogue of Khinchin's theorem in this setting. There have been partial results in this direction, it is my intention to investigate these problems in the setting of self-similar measures.