Regularity and embeddability of self-similar processes

Background. Classical shapes in geometry such as lines, spheres, and rectangles are only rarely found in nature. More common are shapes that share some sort of self-similarity. For example, a mountain is not a pyramid, but rather a collection of mountain-shaped rocks of various sizes down to the size of a gain of sand. Without any sort of scale reference, it is difficult to distinguish a mountain from a ragged hill, a boulder, or ever a small uneven pebble. These shapes are omnipresent in the natural world from clouds to lightning strikes or even trees. What is a tree but a collection of tree-shaped branches? A central component of fractal geometry is the description of how various properties of geometric objects scale with size. Heuristically, measures are mass distributions in space and the study of their geometric properties has been of major interest for many decades. Dimension theory studies scalings by means of various dimensions, each capturing a different characteristic. The most frequent scaling encountered in geometry is exponential scaling (e.g. surface area and volume of cubes and spheres) but even natural measures can simultaneously exhibit very different behaviour on an average scale, fine scale, and coarse scale. Dimensions are used to classify these objects and distinguish them when traditional means, such as cardinality, area, and volume, are no longer appropriate. While abstractly defined, these various dimensions can often be used to classify shapes found in the real world. For instance, the dimension of a porous substance is linked to its chemical reactivity, as is the dimension of a stock ticker to the volatility of a market. Measures arise in many areas of pure and applied mathematics, so having a good understanding of their dimensional properties can lead to answers to seemingly distinct questions in diverse subject areas such as combinatorics, group theory, number theory, coding theory, data processing, and financial mathematics. Research Project. Over the past few years many advanced techniques have been devel- oped to understand such scaling. A drawback of many modern techniques is that they require strong restrictions on the dynamical structure, often in the form of separation conditions that limit how much mass can accumulate in a small space. One way of eliminating these conditions is by considering generic systems; many deep and groundbreaking results were answered this way, even if the general case seems intractable. The benefit of considering generic systems can be attributed to the combination of many small random effects cancelling each other to smooth out the parts that provide difficulty in the rigid general structure. We are interested in quantifying how exactly the randomisation of measures makes them smoother and what structure they share, i.e. we want to determine their regularity. To this end, we will analyse extremal structures in measures arising from dynamical systems, quantify how randomisation of tree structures under random homeomorphisms changes dimensions of subtrees, and study the embeddability of random processes into each other. These extremal structures can be measured by the family of Assouad dimensions, which have only recently gained attention in fractal geometry. Further study of their properties will be central to this project. We will achieve our goal by adapting several modern techniques to random processes and study how they can be optimised, or improved upon, by the randomisation. Our goal is to shed new light on topics such as quantum gravity and percolation problems in graph theory. 1

Self-similarity is a phenomenon where an object looks similar to itself on many scales: The shape of a long snaking river is indistinguishable from that of small tributary streams or creeks. On the small scale this is not a perfect copy of the large-scale object but a random variation of it. These processes and their abstract mathematical counterparts can be referred to as "stochastically self-similar objects" or a "self-similar (stochastic) process". An important example is Brownian motion which models the random path of an object (such as pollen in the air) or the ticker of a publicly traded stock. These processes continue evolving in time, a form of (random) dynamical system. In this project we investigated several such processes from the viewpoint of dimension theory. Dimension theory uses various notions of "dimensions" to quantify irregularity and complexity. One of the main aims of this project was to link those phenomena more closely by studying the "essence" of complicated structures as well as how the complexity of a fixed object changes when its surroundings are changed in a random way, an effect linked to quantum gravity. I highlight the two main achievements of this project. The first involved the analysis of general sets and how their complexity changes under random self-similar changes to the space. A celebrated result from 2008 showed gave an elegant formula for the (Hausdorff) dimension of the randomly changed set in terms of just the dimension of the unchanged set and the randomness used. It was widely believed that the same would hold for the closely related box-counting dimension. However, in joint work with Kenneth Falconer (University of St Andrews) we showed that this is only true if the set is very regular. Otherwise, it is possible for the randomly changed set to have a different dimension. Not only that, but we also showed that it is plainly impossible to give a concise formula for the changed dimension just in terms of the original dimension, one needs to know more about the structure of the set itself. The second project worked on adapting recent advances in Diophantine analysis, that studies how well structures can be approximated, to random attractors. While much was known on the general structure of these objects, no finer analysis was possible with existing tools. In joint work with Simon Baker (University of Birmingham), we adapted Diophantine techniques to the random setting to drastically improve upon old results. These novel results do not just give fine information on the number theoretic properties of self-similar random processes but also naturally extend to "self-affine" processes. These structures have been very difficult to tackle, and we made significant progress towards understanding their fine details.