My Research Interests

This page is very much work in progress, and will likely be more complete when the prospects section of my thesis is finished.

Not to mention the fact that field of Fourier transforms of measures (and related fields) is currently very actively researched; problems are being solved and discovered every year!

We find ourselves in a situation where many questions about the interplay between nonlinear dynamics and Fourier transforms have been answered in recent years, but there are still so many questions yet to be answered.

The case of zero-dimensional measures

In all this work we assume that the dimension of the measure is non-zero. In the zero case, we have less tools at our disposal. In the infinite Lyapunov exponent case, we lose the control from large deviations, which in turn likely means using the sum-product theory of Bourgain will not be possible. This is also the case for the infinite entropy case. Vastly different techniques will need to be employed for these cases. For the zero entropy case, we would start by asking about the existence of such measures.

Everyone's favourite question when considering one-dimensional systems

Asking about higher dimensional analogues is a natural question, and one that many people are interested in. The main thing to note is that the paper of Stoyanov works in higher dimensions. However, the difficulty is likely to be in their application. Li-Pan-Naud [LPN] proved a higher dimensional version of Bourgain-Dyatlov [BD] which took a great deal of work and took full advantage of the strength of the Schottky structure of Fuchsian groups. It might be that without such a structure where the well-distributed behaviour of derivatives is strong, it might not be possible to prove results. However, we said the same thing about extending the Gauss map decay theorems to general nonlinear maps, and that turned out to be possible. I have no doubt that if a higher dimensional analogue will be possible, it will take the commitment of the best mathematicians in this field.

The bridge from nonlinear to linear

So we have Fourier decay results for the case of Markov map invariant dynamics when considering Mosquera-Shmerkin [MS] and Sahlsten-S together. We also have a variety of theorems in the non-homogenous linear setting, such as those presented by Li-Sahlsten [LS1] [LS2]. In their setting, the best decay rate they can achieve is logarithmic, but in the nonlinear and homogenous setting we can get polynomial decay. An important consideration is how the polynomial decay rate of nonlinear systems behaves when you make them more linear. In Solomyak's [S] work, he shows that every self-similar measure has polynomial Fourier decay outside a set of measures with dimension zero. This suggests that the decay results obtained by Li-Sahlsten are not the best possible in most cases. The question is whether proving polynomial Fourier decay for general self-similar measures (linear dynamics) would be possible, or whether the homogeneous property is necessary to use tools like in Mosquera-Shmerkin.

Large Deviations