Workshop on shrinking targets in Hamburg, November 13–15, 2019

Rathaus


Poster


Organisers:
Maxim Kirsebom
Philipp Kunde
Tomas Persson

If you want to participate, please contact the organisers!

The workshop is funded by means of the University of Hamburg's status as a University of Excellence.

Hamburg


Participants

Demi Allen, Bristol
Ekaterina Bagaeva, Nice
Simon Baker, Birmingham
Mats Bylund, Lund
Charis Ganotaki, Lund
Mark Holland, Exeter
Maxim Kirsebom, Hamburg
Simon Kristensen, Århus
Philipp Kunde, Penn State
Georgios Lamprinakis, Lund
Keivan Mallahi-Karai, Bremen
Florian Noethen, Hamburg
Tomas Persson, Lund
Alessandro Pezzoni, York
Sören Schwenker, Hamburg


Location

Hamburg, "Geomatikum".

The department of mathematics is in the building "Geomatikum". Its location can be seen in this map. It is in the building with ''Geologisch-Paläontologisches Museum''.


Program

Wednesday

9:30–10:30 (Room 1240): Registration and coffee
10:30–10:45 (Room 1240): Welcome by the organizers
10:45–11:45 (Room 1240): Simon Baker, ''Overlapping iterated function systems from the perspective of Metric Number Theory''

12–14: Lunch

14:00–15:00 (Room 1111): Mark Holland, ''On almost sure convergence of maxima for dynamical systems''
15:00–15:30: Break
15:30–16:30 (Room 1111): Open problems session


Thursday

9:00–9:30 (Room 1528): Coffee
9:30–10:30 (Room 1528): Tomas Persson, ''Shrinking targets and eventually always hitting points''
10:30–10:45 (Room 1528): Break
10:45–11:45 (Room 1528): Demi Allen, ''Diophantine Approximation on Fractals: Hausdorff Measures of Shrinking Targets on Self-Conformal Sets''

12–14: Lunch

14:00–15:00 (Room 1528): Simon Kristensen, ''Discrepancy, targets and Littlewood type problems''
15:00–15:30 Break
15:30–16:30 (Room 1528): Open problems session


Friday

9:30–10:30 (Room 838): Alessandro Pezzoni, ''Kleinbock–Margulis and points with algebraic-conjugate coordinates''
10:30–11:00: Break
11:00–12:00 (Room 838): Keivan Mallahi-Karai, ''Towards an extreme value law for unipotent flows''

Lunch/departures


Abstracts

Demi Allen, ''Diophantine Approximation on Fractals: Hausdorff Measures of Shrinking Targets on Self-Conformal Sets''

In 2007, Levesley, Salp, and Velani showed that the Hausdorff measure of the set of points in the middle-third Cantor set which can be approximated by triadic rationals at a given rate of approximation satisfies a zero-full dichotomy. More precisely, the Hausdorff measure of the set in question is either zero or full according to, respectively, the convergence or divergence of a certain sum which is dependent on the specified rate of approximation. In this talk, I will discuss an analogue of this result in the setting of more general self-conformal sets satisfying the open set condition. This talk is based on joint work with Balázs Bárány (Budapest).


Simon Baker, ''Overlapping iterated function systems from the perspective of Metric Number Theory''

Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the divergence of naturally occurring volume sums. Importantly this result provides a quantitative description of how the rationals are distributed within the reals. In this talk I will discuss some recent work where I prove that a similar Khintchine like phenomenon occurs typically within many families of overlapping iterated function systems. Families of iterated function systems these results apply to include those arising from Bernoulli convolutions, the 0,1,3 problem, and affine contractions with varying translation parameter. Time permitting I will discuss a particular family of iterated function systems for which we can be more precise. Our analysis of this family shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures.


Mark Holland, ''On almost sure convergence of maxima for dynamical systems''

I will discuss the problem of determining the existence (or otherwise) of an almost sure growth rate for the maximum of a time series of observations, as generated by a dynamical system. In the context of extreme value theory, such results are analogous to having a `strong law of large numbers', where now we look at maxima, rather than sums. Various dynamical system examples will be considered, e.g. hyperbolic systems, or non-uniformly expanding interval maps. The main techniques used to determine the behaviour of maxima involve Strong Borel Cantelli results for certain shrinking target sequences. Given time, I will also mention other methods that can be used (in progress) to establish almost sure behaviour of maxima.


Simon Kristensen, ''Discrepancy, targets and Littlewood type problems''

We present some work originally aimed at studying the Littlewood conjecture, the related Mixed Littlewood conjecture and other related problems via the notion of discrepancy of sequences. Viewed appropriately, the approach is in fact a shrinking target problem, and it is natural to suspect that the approach has uses in a more general framework. The original work is joint with A. Haynes and J. L. Jensen.


Keivan Mallahi-Karai, ''Towards an extreme value law for unipotent flows''

The logarithm law proven by Athreya and Margulis describes the speed of escape of the unipotent flow in the space of lattices for almost every lattice. In a work in progress with Maxim Kirsebom, we will consider the question of determining the probability that the orbit of a random lattices does not run “too fast into the cusp” and establish some results in this direction. The methods used are based on classical ideas from geometry of numbers.


Tomas Persson, ''Shrinking targets and eventually always hitting points''

This talk will be about so called eventually always hitting points. Given a shrinking target, these points are those whose first n iterates will never have empty intersection with the n-th target for sufficiently large n. I will talk about various results on when the set of eventually always hitting points has measure full or zero measure, in particular I will talk about recent results obtain in colaboration with Maxim Kirsebom and Philipp Kunde.


Alessandro Pezzoni, ''Kleinbock–Margulis and points with algebraic-conjugate coordinates''


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Tomas Persson 2019-12-07