Peter Varju

Self-similar measures

Given a finite collection of contractive similarity transformations $f_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,f_n:R^d\to R^d$ and a probability vector $p_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,p_n$, there is a unique probability measure $\mu$ on $R^d$ with the property $\mu =p_1{f}_1(\mu )+\dots <!--\; FUNCTION\; APPLICATION\; -->+p_n{f}_n(\mu )$. Such measures are called self-similar. We will discuss recent developments about the dimension and absolute continuity with respect to Lebesgue measure of these measures.

Jun Luo

On core entropy of polynomials and angle-doubling dynamics

We introduce a universal approach of defining the core entropy for all polynomials of degree at least two. Particularly, in degree two we address the basic properties of the core entropy function. This function is well-defined on the whole Mandelbrot set. Among others, we will show how it is related to the angle-doubling map on the unit circle.

Amir Algom

On random walks and Fourier decay

In recent years there has been an explosion of interest and progress on the Fourier decay problem for fractal measures. In this talk I will discuss some of these results, focusing mainly on how ideas and methods from random walks and spectral theory of transfer operators play a key role in their proof.
Based on joint works with Federico Rodriguez Hertz and Zhiren Wang.

Goran Radunović

On support measures and complex dimensions of fractals

In this talk our goal is to relate and unify two theories each of which give rise to a generalized Steiner formula for arbitrary compact sets. The first one being the Steiner-like formula of Hug, Last and Weil given in terms of the support measures of compact sets and the second one arising from the theory of complex fractal dimensions (by Lapidus and collaborators) and the associated distance zeta function. In its original form, the Steiner-like formula of Hug, Last and Weil may not be very operational if applied to fractal sets. Hence, in order to remedy this and extract finer information for such sets, we introduce and study tightly related geometric functions called the basic functions (of compact sets) and appropriate critical scaling exponents which we relate to the Minkowski dimension of the given set. On the other hand, we show how the distance zeta function decomposes into a sum of Mellin transforms of the newly introduced basic functions which allows one to study them in even finer detail by using the machinery and tools of the theory of complex dimensions. This provides a first step in the unification of these two theories and shows how the complex dimensions are generated by the basic functions. Appropriate functional equations for the corresponding basic zeta functions are also derived which turn out to be very useful in explicit computations. Furthermore, we also comment on the connection to the theory of fractal curvature measures as well as provide interesting examples to support our results.

Ariel Rapaport

Dimension of Bernoulli convolutions in ${ℝ}^d$

Let $({\lambda }_1,...,{\lambda }_d)=\lambda \in {(0,1)}^d$ be with ${\lambda }_1>...>{\lambda }_d$ and let ${\mu }_{\lambda }$ be the distribution of the random vector ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n\ge 0}\pm \left({\lambda }_1^n,...,{\lambda }_d^n\right)$, where the $\text{±}$ are independent fair coin-tosses. Assuming $P({\lambda }_j)\ne 0$ for all $1\le j\le d$ and nonzero polynomials with coefficients $\pm 1,0$, we show that $\dim <!--\; FUNCTION\; APPLICATION\; -->{\mu }_{\lambda }=\min <!--\; FUNCTION\; APPLICATION\; -->\left\{d,{\dim <!--\; FUNCTION\; APPLICATION\; -->}_L{\mu }_{\lambda }\right\}$, where ${\dim <!--\; FUNCTION\; APPLICATION\; -->}_L{\mu }_{\lambda }$ is the Lyapunov dimension. This extends to higher dimensions a result of Varjú from 2018 regarding the dimension of Bernoulli convolutions on the real line. Joint work with Haojie Ren.

Kenneth Falconer

70 Years of Fractal Projections

In 1954 John Marstrand published a paper that included a proof of what is now referred to as Marstrand's Projection Theorem. Since then numerous generalisations, specialisations and variants of this theorem have been developed. The talk will some of these, and present some old and new results on this theme.

Tuomas Sahlsten

$L^2$ flattening in Fourier decay

We present a unified approach to bound the decay rate of Fourier transforms of fractal measures based on the $L^2$ flattening phenomenon in ${ℝ}^d$ together with sufficient separation of random frequencies sampled by the dynamics. The method applies simultaneously to Diophantine self-similar-, non-linear self-conformal- and non-linear non-conformal measures as well as the Patterson-Sullivan measures of convex co-compact hyperbolic manifolds generalising many recent results to higher dimensions and avoiding e.g. renewal theory and the discretised sum-product theorem. Consequences include new equidistribution theorems for vectors on fractals, fractal uncertainty principles and new bounds for the essential spectral gap of scattering resonances on hyperbolic manifolds. Joint work with Simon Baker (Loughborough) and Osama Khalil (Chicago)

Christoph Bandt

Automata-generated compact spaces and the scenery flow of self-similar sets

Neighbor graphs and neighborhood graphs have been used by various authors to study topological and geometrical properties of self-similar and self-affine sets. We consider such finite-state automata in a much more general context including graph-directed constructions, Kleinian limit sets, certain Julia sets, and all simplicial complexes. Our automata describe the quotient map from a symbolic space onto a topological space. In the abstract setting we define the dynamical boundary, which is again an automata-generated space, and the dynamical interior, which is a certain type of manifold. The interior of the Sierpinski triangle, for instance, is the Appolonian gasket with its rich group of homeomorphisms. For finite type self-similar sets this viewpoint leads to new methods of visualization and new parameters for description. A slightly modified version of the scenery flow is given by a Markov chain.

Thomas Jordan

Multifractal analysis for self-affine measures with diagonal matrices

Multifractal analysis typically involves defining level sets in terms of some local quantity and seeing how some suitable notion of dimension (usually Hausdorff). In this talk we will look at this for the local dimension of self-affine measures where the matrices in the system are diagonal. We will show how even very simple examples can yield phase transitions and for the Lalley-Gatzouras family in ${ℝ}^2$ we will look at the problem of whether the level sets satisfy a variational principle. That is whether their Hausdorff dimension is given by the superemum of dimension of Bernoulli measures supported on the set. This is joint work with Istvan Kolossvary and Alex Rutar.

Malabika Pramanik

Numbers - are they normal?

They say the only normal people are the ones you don't know very well. What about numbers? Which ones are normal, and how well do we know them? The notion of mathematical normality is related to the occurrence of different digits in a number. Roughly speaking, a normal number is one in which every block of digits appears with the same limiting frequency. For example, 0.12345678910111213.. is normal in base 10, but 0.1212121212... is not. Normality of numbers is connected to many areas of mathematics, like diophantine approximation, ergodic theory, geometric measure theory, analysis, fractal geometry and computer science. We will discuss a few open problems about normal numbers that lie at the intersection of harmonic analysis and measure theory, and mention some recent progress on them.

Jeremy Tyson

Distortion of dimension by Sobolev and quasisymmetric mappings

We discuss the distortion of metric notions of dimension by mappings in various regularity classes, including mappings with first-order Sobolev regularity as well as homeomorphisms with well-behaved metric distortion (e.g. quasiconformal or quasisymmetric mappings). The study of quasiconformal distortion of Hausdorff dimension originated in the work of Gehring and Väisälä in the 1970s. We discuss Sobolev distortion of Hausdorff dimension as well as other notions (Minkowski dimension, Assouad dimension, and various interpolating quantities such as the Assouad spectrum). For families of subsets indexed by a suitable probability measure, we obtain almost sure dimension distortion estimates which improve on the universal estimates which hold for all source sets of a given dimension. Beyond the classical Euclidean setting, such questions are of interest in other ambient environments such as sub-Riemannian Carnot groups and metric measure spaces satisfying the Poincaré inequality.

Henna Koivusalo

Dynamical subsets in iterated function systems

The classical shrinking target problem concerns the following set-up: Given a dynamical system $(T,X)$ and a sequence of targets $(B_n)\subset X$, we investigate the size of the set of points $x\in X$ for which $T^n(x)\in B_n$ for infinitely many $n$. There are many natural variants of this problem, some of which are much harder than the original shrinking target problem and require a vastly different toolkit. For example, one might be interested in the size of the liminf set of eventually always hitting points; or the size of a dynamical covering set, a limsup set of points hitting infinitely many target sets translated along orbits. In this talk I will discuss some of these problems in the context of iterated function systems, finishing with a hot-off-the-press result on dynamical covering sets (joint with Balazs Barany and Sascha Troscheit).

Hiroki Sumi

Mean stability and bifurcations in random dynamical systems of polynomial automorphisms on ${ℂ}^2$

We consider random dynamical systems of polynomial automorphisms of ${ℂ}^2$ (complex generalized Hénon maps and their conjugate maps) on ${ℂ}^2$. We show that a generic such system has the “mean stability” and satisfies that (1) for every initial value $z$, for almost every sequence of maps in the system, the Lyapunov exponents of the sequence at $z$ are negative, and (2) there exist only finitely many minimal sets of the system and those minimal sets are attracting. Note that those phenomena (1)(2) cannot hold for any deterministic dynamical system of iterations of a single complex generalized Hénon map. To show the above results, we investigate the bifurcations in families of random dynamical systems of polynomial automorphisms on ${ℂ}^2.$ We see many phenomena (so-called randomness-induced phenomena) in random dynamical systems which cannot hold for any deterministic dynamical system of a single map. In this talk we see “randomness-induced order”. In fact, many kinds of maps in one system cooperate together automatically so that they make the chaoticity of the system much weaker and give the system a kind of order. We call such a phenomenon “cooperation principle”. Note that even if the random dynamical system has such an order, the system still has a kind of complexity. Thus it is an interesting problem to observe the gradation between chaos and order for such systems. For the study of complex-one-dimensional random dynamical systems, see H. Sumi, Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton’s Methods, Commun. Math. Phys. 384, 1513–1583 (2021).

Anna Rozanova-Pierrat

Inverse and scattering problems on extension domains

I will present recent results obtained in collaboration with G. Claret, M. Hinz and A. Teplyaev, mostly presented in the preprint https://hal.science/hal-04505158 We use the well-posedness of transmission problems on classes of two-sided Sobolev extension domains to give variational definitions for (boundary) layer potential operators and Neumann-Poincaré operators. These classes of domains contain Lipschitz domains, and also domains with fractal boundaries. Although our variational formulation does not involve any measures on the boundary, we recover the classical results in smooth domains by considering the surface measure on the boundary. We discuss properties of these operators and generalize basic results in imaging beyond the Lipschitz case (see works of H. Ammari, H. Kang and J.K. Seo for the Lipschitz case). In addition, in the case of a bounded extension domain with the outside corkscrew condition, the generalization of the Dirichlet-to-Neumann (or Poincaré-Stek lov) operator allows us to generalize Alessandrini’s (1991) stability result for the Calderon inverse conductivity problem.

Sanju Velani

Shrinking targets and recurrence

Let $(X,d)$ be a compact metric space and $(X,\mathcal{𝒜},\mu ,T)$ be a probability measure preserving system. Furthermore, given a real, positive function $\psi :\mathbb{N}\to {ℝ}_{\ge 0}$ let

denote the associated recurrent set, and given a point $x_0\in X$ let

denote the associated shrinking target set. Under certain mixing properties it known that if ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n\in \mathbb{N}}\psi (n)$ diverges then both the recurrent and shrinking target sets are of full $\mu$-measure. The purpose of this talk is discuss the potential quantitative strengthening of these full measure statements.

This is ongoing work with Bing Li (SCUT) and David Simmons (York).

Li Chen

Heat kernel gradient bounds on the Vicsek set

In this talk, we discuss functional inequalities and gradient bounds for the heat kernel on the Vicsek set. The Vicsek set has both fractal and tree structure, whereas neither analogue of curvature nor obvious differential structure exists. We introduce Sobolev spaces in that setting and prove several characterizations based on a metric, a discretization or a weak gradient approach. We also obtain general $L^p$ Poincaré inequalities and pointwise gradient bounds for the heat kernel. These properties have important applications in harmonic analysis like Sobolev inequalities and the Riesz transform. Moreover, several of our techniques and results apply to more general fractals and trees.

Franklin Mendivil

Generalized Assouad-like dimensions

The Assouad and quasi-Assouad dimensions quantify a set’s most extreme local box-counting behaviour over some (moving) range of scales. This is also true for various refinements of these dimensions such as the $𝜃$-Assouad spectrum and the Assouad-like $\varphi$-dimensions, which provide more specific control over the range of scales considered. These same Assouad-like dimensions are also available to quantify Borel measures. In this talk I will introduce the $\varphi$-dimensions and tell a story of how we used them in the context of cut-out subsets of $[0,1]$ (in particular random rearrangements of them) and then in studying random self-similar Moran constructions.

Ian Morris

Linear images of self-affine fractals

In this talk I will describe how Falconer's classical results on the Hausdorff dimension of self-affine fractals can be extended to the images of those sets under arbitrary linear projections. I will also describe some consequences for the \emph{exceptional projections} of self-affine sets in the sense of Marstrand's projection theorem. This is joint work with Çağrı Sert.

Yilin Wang

The Brownian loop measure on Riemann surfaces and applications to length spectrums

Lawler and Werner introduced the Brownian loop measure on the Riemann sphere in studying Schramm-Loewner evolution. It is a sigma-finite measure on Brownian-type loops, which satisfies conformal invariance and restriction property. We study its generalization on a Riemannian surface $(X,g)$. In particular, we express its total mass in every free homotopy class of closed loops on $X$ as a simple function of the length of the geodesic in the homotopy class for the constant curvature metric conformal to $g$. This identity provides a new tool for studying Riemann surfaces’ length spectrum. One of the applications is a surprising identity between the length spectrums of a compact surface and that of the same surface with an arbitrary number of cusps. This is a joint work with Yuhao Xue (IHES).

Sebastian Andres

Gaussian vs. Sub-Gaussian Behaviour of Random Conductance Models

The random conductance model (RCM) is a well-established model for a random walk in random environment which has been object of intensive study in the last 15 years. One particular question of interest is whether homogenization occurs, in the sense that the deformities in the random medium are ‘smoothed out’, so that only an averaged effect remains. Central probabilistic questions in this context include functional central limit theorems for the random walk or local limit theorems for its heat kernel. On the other hand, there are situations when due to trapping effects the local inhomogeneities of the medium persist in the scaling limit and some kind of anomalous, sub-Gaussian behaviour is seen, which is typically observed for Brownian motions on fractal spaces, for instance. In this talk we will give a detailed review those results. In the last part we will discuss some recent results for RCMs with long-range jumps. This last part is based on a joint work in progress with Martin Slowik (Mannheim).

Sascha Troscheit

Quasi-isometric equivalence of Galton-Watson trees and statistically self-similar sets

The Galton-Watson tree, a fundamental model of random trees, is characterised by a single-root vertex from which each descendant vertex independently generates a random number of offspring, with identical distribution. This model underpins numerous critical phenomena and arises in e.g. the study of stochastically self-similar sets. Drawing on the seminal work of Basu, Sidoravicius, and Sly, which established the almost sure bi-Lipschitz equivalence of two independent percolations on the lattice ${\mathbb{Z}}^d$, we prove the almost sure quasi-isometric equivalence of a large family of Galton-Watson trees. This also implies quasi-symmetric equivalence of a large class of stochastically self-similar sets under some separation conditions. The proof has interesting links to the “car-parking problem” on trees which I will also present in this talk. Based on joint work with Jayadev Athreya.

Martina Zähle

Lipschitz-Killing curvatures for different classes of fractals

In this talk we will present some new results for fractal versions of classical Lipschitz-Killing curvatures, including the Minkowski content as marginal case. In the classical differential geometric context they form a complete system of certain geometric invariants. Recent extensions of former fractal notions and relationships concern a large class of random code-tree based fractals with dependencies as well as domains with deterministic piecewise self-similar boundaries. It turned out that such random fractals, including the V-variable model, show a different stochastic behavior than random recursive constructions. However, the formulas for the mean values obtained by approximation with parallel sets in all cases are structurally the same as those for deterministic self-similar sets. (Joint work with Jan Rataj and Steffen Winter.)