Professional Activities

Jan 1, 2024

Table of Contents

All lists go back to September 2019, prior activities can be found in the CV.

Publications

See the publications page for complete list.

Conference talks / participation

2024

Fractal Geometry and Stochastics 7, 23-27 September 2024, Technical University of Chemnitz, DE.
Talk
Title: Dimension interpolation on planar carpets.
Abstract: There are plenty of sets whose dimension differs depending on the notion of dimension considered. The idea of dimension interpolation is to gain more nuanced geometric understanding of these sets by introducing a one parameter family of dimensions that interpolate in a meaningful way between two wellstudied notions. In the talk I will speak about intermediate dimensions and the Assouad spectrum. Moreover, results will be presented on the exact form of these interpolations as a function of the parameter for planar carpets in the BedfordMcMullen and Gatzouras-Lalley class. Based on joint works with A. Banaji, J.M. Fraser and A. Rutar.
Fractals and Hyperbolic Dynamical Systems, Fall 2024, Alfréd Rényi Institute of Mathematics, HU.
XLIV Dynamics Days Europe, 29 July - 2 August 2024, Constructor University, DE.
CMI-HIMR Summer School on Symmetry and Randomness, 15 - 19 July 2024, University of Bristol, UK.
Workshop on Ergodic Theory and Fractal Geometry, 8-10 July 2024, Loughborough University, UK.
Talk
Title: Assouad spectrum of Gatzouras-Lalley carpets.
Abstract: The talk will focus on the fine local scaling properties of a class of self-affine fractal sets called Gatzouras-Lalley carpets by looking at its Assouad spectrum. This is a one parameter family of dimensions that interpolate between the box and (quasi-)Assouad dimension of the set. We will show a formula for the Assouad spectrum of all Gatzouras-Lalley carpets obtained by taking the concave conjugate of an explicit piecewise-analytic function combined with a simple parameter change. Time permitting, we will demonstrate a number of novel properties that our formula exhibits for dynamically invariant sets and give some hints of the proof strategy. Based on joint work with A. Banaji, J.M. Fraser and A. Rutar.
Geometry and Fractals Under the Midnight Sun, 25-28 June 2024, University of Oulu, FI.

2023

One day ergodic theory meeting, 22 November, University of Warwick, UK.
Dynamics and asymptotics in algebra and number theory, 11-15 September, Bielefeld University, DE.
Recent advances in ergodic theory and dynamics, 13-14 July, Loughborough University, UK.
Fractal Geometry Workshop, 3-7 July, ICMS, UK.
Talk
Title: The Assouad spectrum of Lalley-Gatzouras carpets.
Abstract: Fraser and Yu determined the Assouad spectrum of Bedford-McMullen carpets when they introduced this new dimension spectra in 2018. In the talk, I will highlight interesting new phenomena that the spectrum exhibits in the more general Lalley-Gatzouras class. Based on joint work with Jonathan M. Fraser and Amlan Banaji.
Multifractal analysis and self-similarity, 26-30 June, CIRM, FR.
School and workshop “Dynamics and fractals”; Thermodynamic Formalism: Non-additive Aspects and Related Topics at the Simons Semester, 7-19 May, IMPAN/Bedlewo, PL.
Talk
Title: A variational principle for box counting quantities.
Abstract: The classical variational principle for topological pressure is an essential tool in thermodynamic formalism. The aim of the talk is to extend the framework into a non-conformal setting where we prove a variational principle for an appropriate pressure that is specifically tailored to calculate box counting quantities. Similarities and differences between the variational principles will be highlighted. As an application, we can derive the $L^q$ spectrum of self-affine measures supported on planar carpets and higher dimensional sponges.

2022

Fractals and Related Fields IV (FARF), 5-9 September, Isle de Porquerolles, FR.
Talk
Title: Various dimensions of self-affine measures on sponges.
Abstract: The classical variational principle for topological pressure is an essential tool in thermodynamic formalism. The aim of the talk is to extend the framework into a non-conformal setting where we prove a variational principle for an appropriate pressure that is specifically tailored to calculate box counting quantities. Similarities and differences between the variational principles will be highlighted. As an application, we can derive the $L^q$ spectrum of self-affine measures supported on planar carpets and higher dimensional sponges.
Geometry of Deterministic and Random Fractals, 27 June - 1 July, Budapest University of Technology and Economics, HU.
Rényi 100, 20-23 June, Hungarian Academy of Sciences, HU.
7th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals, 4-8 June, Cornell University, US.
Talk
Title: On the Convergence Rate of the Chaos Game.
Abstract: This talk will address the question: how long does it take for the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting IFS of which we only assume that its lower dimension is positive? We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. As an application, for Bedford-McMullen carpets, we completely characterize the family of probability vectors that minimize the Minkowski dimension of Bernoulli measures. Based on joint work with B. Bárány and N. Jurga.
Workshop on affine and overlapping iterated function systems 2022, 10-12 May, University of Bristol, UK.
Talk
Title: The $L^q$ spectrum of self-affine measures on sponges.
Abstract: In this talk we consider self-affine sponges in $R^d$ which satisfy a separation condition that gives a certain grid alignment to first level cylinder sets. On the plane these are precisely the Lalley-Gatzouras and Bara'nski carpets. We show that the $L^q$ spectrum of any self-affine measure supported on such a sponge satisfies a variational formula for all real values of $q$. Sufficient conditions are given for the formula to have a closed form. In particular, this is always the case for the box dimension of the sponge. The Frostman and box dimension of these measures is also determined. The result is derived from a more general variational principle for calculating box counting quantities on sponges which resembles the Ledrappier-Young formula for Hausdorff dimension and could be of interest to further study in its own right.
Junior Ergodic Theory Meeting, 28-31 March, ICMS, UK.
Talk
Title: Intermediate dimensions of Bedford–McMullen carpets.
Abstract: Intermediate dimensions were recently introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the self-affine Bedford-McMullen carpets are a natural case for investigation, but until now only very rough bounds for their intermediate dimensions have been found. The aim of the talk is to present a precise formula for the full spectrum with some intuition behind it. Moreover, to highlight a surprising connection of the intermediate dimensions to the multifractal spectrum and Lipschitz equivalence of these carpets. Based on joint work with Amlan Banaji.

2021

Only online meetings due to covid.

2020

Fractals and Dynamics, 13-14 January, Bar-Ilan University, IL.

2019

Thermodynamic Formalism: Dynamical Systems, Statistical Properties and their Applications, 9-13 December, CIRM, FR.

Seminar talks

Applications of the method of types in dimension theory.
Analysis Seminar at the Rényi Institute; April 18, 2024.
Abstract
The method of types stems from information theory and is a basic tool in large deviation theory involving finite alphabets. This approach has recently found its way to proving results in dimension theory, mainly related to the box dimension of sets and measures. The talk will survey some of these results and the argument itself will be demonstrated on the simplest possible example of self-similar sets.
Interpolating between different notions of dimension. Dynamical Systems Seminar at the Jagiellonian University in Cracow; March 22, 2024.
Abstract
Various notions of dimension capture how efficiently a set can be covered from different aspects. These can lead to different values for a set which is “inhomogeneous” in some sense. A recent trend in dimension theory has been to define one parameter families of dimensions which interpolate between two well-known dimensions in order to uncover finer geometric scaling properties of such sets. The talk will present how this can be done in a meaningful way and present results and applications for the class of planar self-affine carpets. Based on joint works with A. Banaji, J.M. Fraser and A. Rutar.
Assouad spectrum of planar carpets.
BudWiSer - The Budapest - Wien Dynamics Seminar; January 12, 2024.
Abstract
There are plenty of examples of sets, such as planar carpets, whose box and Assouad dimensions are different. The Assouad spectrum is a one parameter family of dimensions which gives additional information about the fine scale geometry of such sets. In the talk, we will present a formula for the Assouad spectrum of Gatzouras-Lalley carpets. We will highlight why it can be expressed as the concave conjugate of an explicit piecewise-analytic function and, among dynamically invariant sets, what novel features does the spectrum exhibit. Based on joint work with A. Banaji, J.M. Fraser and A. Rutar.
On the convergence rate of the chaos game.
Universit"{a}t Bremen Dynamical Systems and Geometry Seminar; December 1, 2023.
Abstract
The chaos game is a simple random iterative procedure that generates the attractor of an iterated function system (IFS). This talk will address the question: given a measure driving the chaos game, how long does it take for the orbit of a point to reach a certain density inside the attractor? We show under mild assumptions on the IFS and the measure that the rate of growth of this cover time is determined by the Minkowski dimension of the measure. The results will be illustrated with examples, particularly in finding the measure which achieves the fastest possible rate of convergence. Based on joint work with Balázs Bárány and Natalia Jurga.
Some variants of the variational principle.
University of St Andrews Analysis Seminar; September 26, 2023.
Abstract
A common theme in my recent projects has been to solve optimisation problems usually involving entropies and Lyapunov exponents. In each case the goal is to show a kind of variational principle. I will survey these problems over probability vectors from easier to more complex and show how they come up in the study of Lq dimensions, multifractal analysis and the Assouad spectrum.
Assouad spectrum beyond Bedford-McMullen carpets.
University of St Andrews Analysis Seminar; March 7, 2023.
Abstract
Fraser and Yu introduced the Assouad spectrum in 2018, one of the examples for which they calculated the spectrum were Bedford-McMullen carpets. Natural directions for generalisation are either to consider more general carpets on the plane or try higher dimensional sponges. The talk will focus on how new phenomena lead to additional phase transitions in the spectrum of these more general constructions compared to the single one for Bedford-McMullen carpets. Based on work in progress with Jonathan M. Fraser and Amlan Banaji.
A variational principle for box counting quantities.
University of Warwick Ergodic Theory and Dynamical Systems Seminar; December 6, 2022.
Abstract
The classical variational principle for topological pressure is an essential tool in thermodynamic formalism. The aim of the talk is to extend the framework into a non-conformal setting where we prove a variational principle for an appropriate pressure that is specifically tailored to calculate box counting quantities. Similarities and differences between the variational principles will be highlighted. As an application, we can derive the Lq spectrum of self-affine measures supported on planar carpets and higher dimensional sponges.
Distance between natural numbers based on their prime signature.
University of Warwick Number Theory Seminar; December 5, 2022.
Abstract
One can define different metrics between natural numbers based on their unique prime signature. Fixing such a metric, we are interested in the asymptotic growth rate of the arithmetic function $L(N)$ which tabulates the cumulative sum of distances between consecutive natural numbers up to $N$. In particular, choosing the maximum norm, we will show that the limit of $L(N)/N$ exists and is equal to the expected value of a certain random variable. We also demonstrate that prime gaps exhibit a richer structure than on the traditional number line and pose a number of problems. Joint work with István B. Kolossváry.
Multifractal analysis of self-affine measures on a simple Bara'nski carpet.
University of St Andrews Analysis Seminar; October 4, 2022.
Abstract
Baranski carpets exhibit interesting phenomena not witnessed by systems satisfying some sort of coordinate ordering property. We demonstrate that this is also true for multifractal analysis by looking at self-affine measures on a simple Baranski carpet. Namely, the multifractal formalism fails (i.e. the Legendre transform of the Lq spectrum is not equal to the multifractal spectrum), even though the carpet has no overlaps and its Hausdorff and box dimensions are equal. The spectrum even has a jump discontinuity in case of the natural measure.
The Assouad dimension of self-affine measures on sponges.
University of St Andrews Analysis Seminar; April 26, 2022.
Abstract
A class of self-affine sponges generated by diagonal matrices is introduced which generalise well-known planar constructions to higher dimensions. We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures supported on these sponges. The upper and lower bounds always coincide in dimensions $d=2,3$ yielding precise explicit formulae for the dimensions. Moreover, there are easy to check conditions guaranteeing that the bounds coincide for $d \geq 4$. An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. Joint work with J. M. Fraser.
On the convergence rate of the chaos game.
University of St Andrews Analysis Seminar; October 26, 2021.
Abstract
In the 1988 textbook “Fractals Everywhere”, M. Barnsley introduced an iterative random procedure for generating fractals which is coined the “Chaos Game”. Two natural questions are: what is the expected time taken by this procedure to become r-dense in the fractal, and for which measure can this expected time be minimised? In this talk we will discuss how the box dimension of the measure comes into play and characterise the family of probability vectors that minimise the expected time in the case of Bernoulli measures defined on Bedford-McMullen carpets. Based on joint work with Balázs Bárány and Natalia Jurga.
Calculating box dimension with the method of types.
University of St Andrews Analysis Seminar; February 9, 2021.
Abstract
This talk will present an argument based on using the method of types to calculate the box dimension of sets. Demonstrating first on self-similar sets, we then use it to generalize the formula for the box dimension of self-affine carpets of Gatzouras-Lalley and of Barański type to their higher dimensional sponge analogues. In addition to a closed form, we also obtain a variational formula which resembles the Ledrappier-Young formula for Hausdorff dimension.
Intermediate dimensions of Bedford–McMullen carpets.
Universitat Wien Egodic Theory Seminar; November 12, 2020.
Abstract
The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim_{\theta} \Lambda$, interpolate between its Hausdorff and box dimensions using the parameter $\theta \in [0,1]$. Determining a precise formula for $\dim_{\theta} \Lambda$ is particularly challenging when $\Lambda$ is a Bedford-McMullen carpet with distinct Hausdorff and box dimension. In this talk, after giving an overview on dimension interpolation, we will present an argument that shows that $\dim_{\theta} \Lambda$ is strictly less than the box dimension of $\Lambda$ for every $\theta < 1$. Time permitting, we will also show how to improve on the lower bound obtained by Falconer, Fraser, and Kempton.
Intermediate dimensions of Bedford–McMullen carpets.
University of St Andrews Analysis Seminar; August 11, 2020.
Abstract
The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim_{\theta} \Lambda$, interpolate between its Hausdorff and box dimensions using the parameter $\theta \in [0,1]$. Determining a precise formula for $\dim_{\theta} \Lambda$ is particularly challenging when $\Lambda$ is a Bedford-McMullen carpet with distinct Hausdorff and box dimension. In this talk, after giving an overview on dimension interpolation, we will present an argument that shows that $\dim_{\theta} \Lambda$ is strictly less than the box dimension of $\Lambda$ for every $\theta < 1$. Time permitting, we will also show how to improve on the lower bound obtained by Falconer, Fraser, and Kempton.
Generic self-similar and self-conformal attractors on the line.
University of St Andrews Pure Maths Colloquium; March 5, 2020.
Abstract
In 2001, Peres, Simon and Solomyak considered one-parameter families of self-similar Iterated Function Systems (IFSs) on the line satisfying the so-called transversality condition. They proved that if the similarity dimension is less than 1, then for a typical parameter (both in category and measure sense) the existence of overlaps between cylinders implies that the appropriate dimensional Hausdorff measure of the attractor is zero. We extend this result both for self-similar and self-conformal IFSs on the line. Moreover, combining with recent results of Fraser-Henderson-Olson-Robinson and Angelevska-Kaenmaki-Troscheit we obtain that the Assouad dimension of such systems is 1. (joint work with Balázs Bárány, Mihal Rams, and Károly Simon)
Typical self-similar and self-conformal attractors on the line.
University of Bristol Ergodic Theory and Dynamical Systems Seminar; February 27, 2020.
Abstract
In 2001, Peres, Simon and Solomyak considered one-parameter families of self-similar Iterated Function Systems (IFSs) on the line satisfying the so-called transversality condition. They proved that if the similarity dimension is less than 1, then for a typical parameter (both in category and measure sense) the existence of overlaps between cylinders implies that the appropriate dimensional Hausdorff measure of the attractor is zero. We extend this result both for self-similar and self-conformal IFSs on the line. Moreover, combining with recent results of Fraser-Henderson-Olson-Robinson and Angelevska-Kaenmaki-Troscheit we obtain that the Assouad dimension of such systems is 1. (joint work with Balázs Bárány, Mihal Rams, and Károly Simon)
Counting intersections in transversally overlapping self-affine planar carpets.
University of St Andrews Analysis Seminar; November 5, 2019.
Abstract
The first part of the talk will give a broader overview of where this joint work with Károly Simon fits into the literature about self-affine planar carpets. Then more detail will be given about the result for box dimension. In particular, the key argument for counting intersecting boxes under a certain transversality condition will be presented.
Overlapping self-affine planar carpets.
University of Manchester Analysis and Dynamics Seminar; October 21, 2019.
Abstract
We will survey the different types of self-affine carpets in the literature that led Károly Simon and myself to study a general class of carpets with different types of overlapping cylinders. These overlaps could cause the dimension of the attractor to drop compared to an analogous system without overlaps. We showed explicitly checkable conditions on the parameters of the carpet, under which this (possible) drop of different notions of dimension does not occur. The results will be illustrated with several examples and an application to three-dimensional systems. Time permitting, I will give some ideas on how we handled overlaps in the proofs.