Overview

My work is currently centered to pure mathematics but it has many intradisciplinary features. I am interested of working on multiple fields such as analysis, arithmetics, geometry, dynamics and stochastics and exploring new connections between them. The center of this goal over the past years has been the theory of dynamical systems and applications in other fields. For example, I am currently focused on the links between harmonic analysis, dynamics and stochastics in the context of fractal geometry and quantum mechanics. Moreover, I am developing the theory of the scenery flow and Furstenberg's CP-chains which have proven to have some deep connections to classical arithmetic and geometric problems and I have worked on developing these connections.

In the following I will give an exposition of my past work and the list of publications. If there are some errors or you want to make suggestions, do not hesitate to contact me. The preprints of the publications are hosted on arXiv where one can find hopefully the most updated versions.

Equidistribution of eigenfunctions of the Laplacian

Quantum chaos is a field that is aimed to study the properties of eigenfunctions of the Laplacian (stationary quantum states) on a Riemannian manifold using the chaotic properties of the underlying geodesic flow in high energy, that is, in the large eigenvalue limit. In this sense the field is connecting quantum mechanics to classical mechanics. A key result in the field is the Quantum Ergodicity Theorem of Shnirelman, Zelditch and Colin de Verdière, which is an equidistribution result of the eigenfunctions for large eigenvalues when the geodesic flow is ergodic. In our recent work we have been attempting to study the theory for a problem of large scale quantum ergodicity, where instead of large energy, we fix an energy window and vary the geometric properties of the manifold.


Dynamics and stochastics in Fourier analysis

The high-frequency asymptotics of Fourier coefficients of functions and measures describe their local structure. For example, they can be used to yield geometric or arithmetic features of the object under study such as on dimension, curvature, equidistribution or combinatoric structure. In my recent work I have been working on finding conditions based on ergodic theory, dynamical systems and stochastics which yield efficient estimates for Fourier transforms of dynamically or randomly constructed objects. Furthermore, we aim to use these estimates to obtain new arithmetic/geometric applications for them.


Scenery flow, CP chains, local entropy and applications

After the recent influential works of Furstenberg and Hochman-Shmerkin techniques based on magnifying measures (and taking their tangent measures) have been essential in the study of arithmetic and geometric features of sets and measures in new settings. For example, these approaches work well with notions that involve questions on the entropy or dimensions of a measure, projections and distance sets, or features related to equidistribution. The key ideas are based on the dynamics or stochastics of the process of magnification and applying classical tools from ergodic theory and Markov chains. I have been recently working on developing these techniques with new arithmetic and geometric applications in mind.


Hyperbolic dynamics in non-compact spaces

Hyperbolic dynamics in non-compact spaces present many new challenges which do not appear in the classical compact hyperbolic setting. For example, natural absolutely continuous invariant measures can be infinite and there can be escape of mass to cusps at infinity. Studying these these dynamical systems is important as various hyperbolic dynamical systems on non-compact spaces can be used to model questions in fields like Diophantine approximation and intermittent transition to chaos. I have recently been working on in particular the countable Markov shift, which provides a natural symbolic model for many non-compact dynamical systems such as countable Markov maps.


  • Pointwise perturbations of countable Markov maps
    with Thomas Jordan and Sara Munday
    Preprint (2016)
    Given an expanding countable Markov map (for example the Gauss map, Lüroth map, induced map for the intermittent Manneville-Pomeau system) we study the topological stability of the map when a small pointwise perturbation is applied. In particular, we study the singularity of the topological conjugacy between the original countable Markov map and the perturbed map. Often the topological conjugacy is a singular function (derivative is zero a.e.) and for many examples their properties are well-studied, most famously for the Minkowski’s question mark function, which is a topological conjugacy between the Gauss map and a Lüroth map. Due to the non-compact nature of countable Markov maps, many standard notions of singularity of the conjugacy behave discontinuously under perturbations. This happens for example for the Hölder exponent of the conjugacy or Hausdorff dimension of the pullback of the absolutely continuous invariant measure under the conjugacy, which can attain any possible values no matter how small the perturbation is. However, we found out that the Hausdorff dimension of the set of points where the derivative of the conjugacy is not zero instead does behave continuously under pointwise perturbations given suitable regularity assumptions on the maps. Using this result we also established a continuity theorem in non-uniformly hyperbolic dynamics for topological conjugacies between Manneville-Pomeau maps under perturbations.

Lorenz maps, kneading theory and β-shifts

Lorenz maps are increasing interval maps with a single discontinuity and they arise naturally as return maps for the dynamics of the Lorenz system, which is classically used to model problems in hydrodynamics. Kneading theory is a typical tool to study such maps and here the dynamics can be understood with the properties of intermediate β-transformations, which with suitable parameters are topologically conjugated to Lorenz maps. I have been recently and currently working on the symbolic model for these maps and studying the relations between the arithmetic properties of β, the kneading invariants and the dynamics of the symbolic model.


  • Intermediate β-shifts of finite type
    with Bing Li and Tony Samuel
    Discrete Contin. Dyn. Syst. Ser. A (2016) Vol 36 (1): 323-344
    When expressing a real number in a non-integer number basis β, it is possible that there are as many as continuum different expansions. As with binary and other expansions, there are natural dynamical systems that can be used to generate the digits and these systems are far from unique. Classical examples are given by the lazy-, greedy- and intermediate β-transformations introduced by Parry. In this work we provide a characterisation of the subshifts of finite type (SFT) property of the β-shift spaces generated by the intermediate β-transformations with respect to the periodicity of the orbits of the kneading invariants, which provides an analogue of a similar characterisation by Parry for the greedy and lazy systems. As an application, we highlight how dynamically different β-transformations can be for a certain fixed β by showing that the corresponding symbolic systems arising from the intermediate β-transformations can be SFT where as the corresponding lazy and greedy ones are not. Moreover, a similar property also holds for the transitivity of the systems.

Tangent measures, singularity and rectifiability

After the seminal work of Preiss the notion of tangent measure has become influential in the study and characterisations of the smoothness properties of sets and measures of integer dimension. A typical way to describe the smoothness is to use rectifiable sets and measures, which approximately behave like smooth manifolds. Rectifiability can be characterised by the flatness features of tangent measures by the classical works of Marstrand and Mattila under suitable density assumptions, and moreover a deeper characterisation can be done merely by the existence of densities by Preiss's theorem. During my PhD studies I mostly worked on the relationships of tangent measures to other smoothness/singularity features of measures and also studied geometric features of rectifiable sets.


  • Tangent measures of typical measures
    Real Anal. Exchange (2015) Vol 40 (1): 53-80
    A surprising construction of O'Neil shows that it is possible to find a measure on the Euclidean space with all Radon measures as tangent measures almost everywhere. In this paper we prove that a generic measure with respect to Baire category in fact satisfies this property of O'Neil. When submitting this paper to a journal we found out that O'Neil himself actually proved the same result in his PhD thesis from 1994 but never published that particular result in a journal, so this paper was prepared while being unaware of the earlier work. The proof is nevertheless slightly different from O'Neil's thesis as our approach is based on dyadic cube filtrations. Our method also applies to prove a new analogous result for measures supported on binary trees and their micromeasures in the sense of Furstenberg.
  • Tangent measures of non-doubling measures
    with Tuomas Orponen
    Math. Proc. Camb. Phil. Soc. (2012) Vol 152: 555-569
    Tangent measures are often used to characterise fractal and smoothness features of measures such as the Marstrand-Mattila theorem characterising rectifiable measures. However, usually these characterisations require an additional density assumption to work. For example, the constructions of Preiss and Freedman-Pitman exhibit that singular measures on the real line can have all tangent measures as constant multiples of Lebesgue measure. In this paper we study a higher degree of singularity, the non-doubling of measures, and analogues of the works of Preiss and Freedman-Pitman for non-doubling measures. We construct a non-doubling measure with every tangent measure as equivalent to Lebesgue measure, which is the best one can hope in this setting. As an application this prevents the characterisation of upper porosity for non-doubling measures with respect to tangent measures, which was left open by a result of Mera-Morán for doubling measures.
  • Radial projections of rectifiable sets
    with Tuomas Orponen
    Ann. Acad. Sci. Fenn. Math. (2011) Vol 36: 677-681
    The Besicovitch-Federer projection theorem is an important result in geometric measure theory that characterises rectifiable sets and measures in the terms of their random orthogonal projections onto hyperplanes. A possible analogue to this characterisation concerns radial projections, which is a long-standing open problem as the parameter space for radial projetions can be a highly singular set. In this paper we prove the 'easy direction' of the analogue of Besicovitch-Federer for radial projections. Moreover, when describing the null exceptional set for the projection theorem in the non-trivial case, we found out that it is always contained in a plane of co-dimension at least two. In two dimensions this means that the exceptional set is at most one point.

Theses

Collaborators

Here are some people I have collaborated or I am currently engaging collaboration with: De-Jun Feng, Andrew Ferguson, Jonathan Fraser, Jasun Gong, Alan Haynes, Mike Hochman, Timo Jolivet, Thomas Jordan, Charlene Kalle, Tom Kempton, Georgie Knight, Vadim Kulikov, Antti Käenmäki, Etienne Le Masson, Jaakko Lehtomaa, Bing Li, Chao Ma, Niko Marola, Sara Munday, Tuomas Orponen, Tony Samuel, Pablo Shmerkin, Ville Suomala and Jim Tseng. Check out their work as well!
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