Dynamics and measures on fractals

Context of research: Fractal geometry is still a relatively young field of mathematics. Due to improvements in computing power, it was fashionable in the popular culture of the 1980s, largely thanks to the work of Mandelbrot. It deals with questions concerning sets that have a complicated structure on fine scales, offering tools for classifying and producing them. Fractal sets can be crudely classified using different notions of dimension, of which we by now have many. When it comes to ways of producing fractals, one of the primary methods is dynamical, through iterated function systems. Despite seemingly straightforward definitions, understanding the dimension theory of fractal sets requires, not just observing the connection to, but also developing the theory of, dynamical systems and geometric measure theory. For a few decades, despite continuous activity, only incremental improvements were made in the methodology of fractal geometry. But in recent years, new players have entered the field, and significant steps have been taken by Barany, Feng, Hochman, Morris, Shmerkin and Rapaport, to name a few. Not only have they proved fundamental new results but, more importantly, in the proofs an array of new techniques from dynamics have been introduced. The most significant improvements have been in the overlapping and non-conformal cases.

Through a classical theorem of Hutchinson we know that for any iterated function system (IFS) (a collection of contractive bijections) a related unique, non-empty, compact, invariant set F exists. The invariant set F usually has fractal features and, indeed, IFSs are the primary means for constructing fractal sets. A central problem in fractal geometry is to express the dimension of the invariant set F in terms of its IFS. Furthermore, an IFS with an invariant set F gives rise to an expanding dynamical system on F, so that questions from classical theory of dynamical systems naturally guide and inspire research in fractal geometry.

Aims, objectives and methodology: Affine IFSs are a natural context for studying recurrence in non-conformal dynamical systems. As described above, dramatic progress has been made in the study of self-affine sets, and it is timely to capitalise on these findings in the context of dynamics on self-affine fractals. The overarching goal of this research project is to identify and solve recurrence problems for dynamical systems on self-affine sets. To give just one example of a specific recurrence problem, we mention the shrinking target problem, introduced by Hill and Velani in the context of Diophantine approximation: Given a sequence of subsets (U_n), how big is the shrinking target set of points x for which T^n(x) belongs to U_n for infinitely many n? Sets of well-approximable points in Diophantine approximation can be interpreted as shrinking target sets. The shrinking target problem has been studied on generic self-affine sets by Barany and Troscheit, and earlier, Koivusalo and Ramirez, but the new techniques of dimension theory provide a stepping stone to improve upon these
by providing results for fixed sets instead of generic ones. We will find a way to adapt these techniques
to the study of dimensions of subsets of self-affine sets. Problems of intermediate difficulty include combining path-dependent and moving target problems in the generic case, and studying recurrence to geometric balls.

Collaborators: We are currently relying on the expertise within the department in making progress in these problems. However, if it turns out to be necessary, we will reach out to the external experts in affine dimension theory.

This project falls within the EPSRC Mathematical Sciences research area.