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.
- Quantum ergodicity and
Benjamini-Schramm convergence of hyperbolic surfaces
with Etienne
Le Masson
Duke
Math. J. (2017), to appear
We present a quantum ergodicity theorem for fixed spectral window
and sequences of compact hyperbolic surfaces converging to the
hyperbolic plane in the sense of Benjamini and Schramm. This
addresses a question posed by Colin de Verdière. Our theorem is
inspired by
results for
eigenfunctions on large regular graphs by Anantharaman and Le Masson. It applies in particular to eigenfunctions on
compact arithmetic surfaces in the level aspect, which connects it
to a question of
Nelson on Maass forms. The proof is based on a wave
propagation approach recently
considered by Brooks, Le
Masson and Lindenstrauss on discrete graphs. It does
not use any microlocal analysis, making it quite different from the
usual proof of quantum ergodicity in the large eigenvalue
limit. Moreover, we replace the wave propagator with renormalised
averaging operators over discs, which simplifies the analysis and
allows us to make use of a general ergodic theorem of
Nevo. As a consequence of this approach, we only require bounded measurable observables.
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.
- On the Fourier analytic structure of the Brownian graph
with Jonathan Fraser
Preprint (2015)
This work continues the previous
article
with T. Orponen where we established that almost surely the graph of the Wiener process (Brownian motion)
cannot be a Salem set answering in part a question Kahane (1993). The proof there was based on an universal
upper bound for Fourier dimension we established with geometric
measure theoretic means and thus it was still unknown what is the
actual value of the Fourier dimension of the Brownian graph and is
it even positive. In this article we present a new method based on
Itō
calculus, which we combine with the classical work of Kahane on Brownian images
(
J-P. Kahane, "Images browniennes des ensembles
parfaits". C. R. Acad. Sci. Paris 263 (1966), 613-615),
to establish that the Fourier dimension of the Brownian graph is
almost surely 1. We also discuss about some consequences of the
decay of Fourier transform for the equidistribution of orbits under
toral endomorphisms.
- Fourier transforms of Gibbs
measures for the Gauss map
with Thomas
Jordan
Math. Ann.
(2016) Vol 364 (3): 983-1023
The decay of Fourier transform at infinity is usually achieved for
random measures, measures with a suitable convolution form or with
some geometric curvature properties. In this paper we study how
dynamical and arithmetic properties of the measure affect its
Fourier transform. We restrict our attention to measures that are invariant
Gibbs measures for the Gauss map. This dynamical system is highly arithmetic as it
describes the evolution of the continued
fraction coefficients of irrational numbers. Moreover, the Gibbs
property provides strong statistical information on the decay of
correlations for the invariant measure. We employ the ideas developed in the
construction of the
Kaufman measure on badly approximable numbers to
prove that the Fourier transform of any Gibbs measure for the Gauss
map with thin enough tail at the cusp and Hausdorff dimension greater than a
half has a power decay. This can be applied to Fourier analysis of
singular monotonic functions and as a
special case this solves
Salem's
problem from 1943 on the Fourier-Stieltjes coefficients of the
Minkowski's question mark function as the corresponding Stieltjes measure is
a Gibbs measure satisfying our assumptions. Moreover, the main
result also applies to the
Hausdorff measure on the
N-badly approximable numbers and with the
Davenport-Erdős-LeVeque theorem we extend some results of
Hochman-Shmerkin on equidistribution properties of Gibbs measures.
- On Fourier
analytic properties of graphs
with Jonathan Fraser and Tuomas
Orponen
Int. Math. Res. Not. IMRN (2014) Vol 2014
(10): 2730-2745
The Fourier dimension of a set describes the optimal rate of decay for the
Fourier transforms of measures supported on the set and it has some deep
arithmetic and geometric interpretations. If the Fourier
dimension equals to Hausdorff dimension, then the set is called a
Salem set. In this article we study the Fourier dimensions of graphs
of continuous functions on the real line and what values the
dimension can generically have. Our main result is that the Fourier
dimension can never exceed the value one, which we base on a Fourier
analytic version of Marstrand's slicing theorem. As a
consequence we answer in negative to Kahane's problem from 1993
(J-P. Kahane. ''Fractals and random measures''. Bull. Sci. Math. (2)
117 (1993), 153-159) where he
asked whether the graphs of natural random functions, such as Brownian
motion, are Salem sets almost surely. We also study the notion of
genericity of continuous functions with respect to Baire category
and prove that with respect to this genericity there can be no
measures supported on the graph with Fourier transform decaying to
zero at infinity, in particular yielding that the Fourier dimension is zero.
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.
- Dynamics of the scenery flow and
geometry of measures
with Antti
Käenmäki and Pablo
Shmerkin
Proc. London Math. Soc.
(2015) Vol 110 (5): 1248-1280
In this paper we apply the ergodic theoretic tools developed for the
scenery flow to classical questions on local distribution of
measures in geometric measure theory (that a priori do not
involve any dynamics). The carrying idea is that the scenery flow
allows to reduce measure
theoretical questions to usually set theoretical
analogues which are often almost trivial. The machinery provides very
short and straightforward abstract proofs for earlier results using the
scenery flow theory as a black box and allows us
to provide new extensions and sharpenings of conical density
theorems and of results in porosity theory. In particular, we provide sharp
constants in Marstrand's conical density theorem and explain the
result in the terms of rectifiability of tangent measures. Moreover,
we prove that the maximal dimension of porous measures is obtained
by the maximal dimension of porous sets, which allows us to reduce
the measure theoretical theory of porosity to its set theoretical analogue.
- Structure of distributions
generated by the scenery flow
with Antti
Käenmäki and Pablo
Shmerkin
J. London Math. Soc.
(2015) Vol 91 (2): 464-494
The scenery flow is a measure valued dynamical system that is supported on the
tangent measures of a measure, where the dynamics is given by
magnification to a fixed point. The ergodic theory for the scenery
flow can be developed to a great extent using Furstenberg's
machinery
of CP chains, which are the symbolic analogues of the
scenery flow. This was in particular done in the work of
Hochman,
where a key finding was that the tangent distributions (empirical
measures for the scenery flow) are
fractal distributions
almost everywhere. Fractal distributions are a special class of
distributions of measures that satisfy pleasant
statistical invariance properties under magnification and translation. In
this paper we provide extensions to the work of Hochman. In
particular, we prove that the set of fractal distributions is closed and forms a Choquet simplex known as the
Poulsen simplex. We apply these results to prove a converse
to the result of Hochman by showing that for any fractal
distribution one can construct a measure generating it and that a Baire
generic measure has every fractal distribution as a tangent
distribution almost everywhere and similarly for CP chains, thus providing a natural analogue to
the result of
O'Neil.
- Scaling scenery of
(×m,×n) invariant measures
with Andrew Ferguson and Jonathan Fraser
Adv.
Math. (2015) Vol 268: 564-602
In this paper we study the applicability of the theory of
Furstenberg
and
Hochman-Shmerkin on CP chains
in non-conformal dynamics. We study the hyperbolic toral endomorphism
that is obtained by multiplying each coordinate by different natural
numbers
m and
n. We prove that if we take an invariant measure for
the system which satisfies strong enough statistical independence on
the natural Markov partition, in our case Bernoulli property, the measure generates an ergodic CP chain. The theory
of Hochman-Shmerkin allows us to give then a sharpening of the
Marstrand's projection theorem for these Bernoulli
measures. Moreover, we present a criterion based on CP chains for the validity of dimension version of
Falconer's distance set conjecture and apply our main result on (×
m,×
n) to
prove the conjecture for Bedford-McMullen
carpets and in a weaker form for Barański and Lalley-Gatzouras
carpets.
- Dimension,
entropy and the local distribution of measures
with Pablo
Shmerkin and Ville
Suomala
J. London Math. Soc. (2013) Vol 87 (1): 247-268
Local entropy averages as considered by
Llorente-Nicolau and
Hochman-Shmerkin is a way to present Hausdorff and packing
dimensions of measures in the terms of their entropy averages over
dyadic cube filtrations. In this article we present a general framework based on the
local entropy averages to prove results in classical questions in
local distributions of measures in geometric measure theory. In
particular, we give the correct mean version of Marstrand's conical
density theorem, an asymptotically sharp dimension bound for mean
k-porous measures and a mean version of the local homogeneity
theorem, which provide natural extensions to several earlier works in
the field.
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
- Fine
Structure of Measures (pdf)
PhD Thesis, University of Helsinki (2012), lectio slides (100 MB!)
This thesis presents the results published in the papers
(a),
(b) and
(c) and gives a historical
overview of their topics.
- Huokoisuus ja dimensiot (pdf)
MSc Thesis (in Finnish), University of Helsinki (2009)
In this thesis we review the notions of fractal dimensions and
their connection to porosity. Most emphasis is given to a
presentation of a
detailed proof of
the main result of
Järvenpää et
al. on the Minkowski dimension of
k-porous sets in
the case
k = 1.
- Huokoiset
joukot (pdf)
BSc Thesis (in Finnish), University of Helsinki (2008)
We present the notion of porosity of sets, study its
basic topological, measure theoretical and dimensional properties,
and give some examples.
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!