### Programme

All talks will be held in LT2 of the chemistry building at the University of Bristol. The entrance to the building is: ///shout.intent.unwanted, at the end of Cantock's Close.

Schedule LMS Scheme 9 - New appointments - 10 May
Tuesday, 10 May Speaker/Activity Title
13:30–14:00 Registration/Welcome In foyer
14:00–15:00 Henna Koivusalo Impressions on dimensions in iterated function systems
15:00–16:00 Tea/Coffee In foyer
16:00–17:00 Tom Kempton Structure in Overlaps of Self-Similar sets and measures.
17:00–18:00 Karma Dajani Random $$\beta$$-transformation on fat Sierpinski gasket
19:00– Workshop Dinner Al Bacio Restaurant

Schedule Workshop on overlapping and affine iterated function systems - 11 & 12 May
Wednesday, 11 May Speaker/Activity Title
09:00–10:00 Charlene Kalle Linear expanding and contracting iterated function systems
10:00–11:00 Mike Whittaker Self-similarity of substitution tiling semigroups
11:00–11:30 Coffee/Tea In foyer
11:30–12:30 Han Yu Absolute continuity of projections of Bernoulli measures on carpets and sponges
12:30–14:00 Lunch
14:00–15:00 Simon Baker Overlapping iterated function systems from the perspective of Metric Number Theory
slides
15:00–16:00 Coffe/Tea In foyer
16:00–17:00 Discussion
17:00–18:15 Lightning Talks Schedule listed below
18:15– Reception & Posters Poster listing below
Thursday, 12 May Speaker/Activity Title
09:00–10:00 Antti Käenmäki Finer geometry of planar self-affine sets.
10:00–11:00 Çağrı Sert Projective fibred fractals.
11:00–11:30 Coffee/Tea In foyer
11:30–12:30 István Kolossváry The $$L^q$$ spectrum of self-affine measures on sponges.
12:30–14:00 Lunch In foyer
14:00–15:00 Natalia Jurga Non-existence of the box dimension for dynamically invariant sets
15:00–15:30 Coffe/Tea In foyer
15:30–16:30 Balázs Bárány Recurrence rates for shifts of finite type

Lightning Talk Schedule
Speaker Title
Roope Anttila Assouad dimension of Invariant Measures for Place Dependent Probabilities
George Bender Intersections of Okamoto's function
Benthen Zeegers Critical intermittency in random interval maps
slides
Vilma Orgoványi Orthogonal projections of the random Menger sponge
poster and slides.
Amlan Banaji Intermediate dimensions of Bedford-McMullen carpets
Aleksi Pyörälä Normal numbers in self-conformal sets
Adam Śpiewak Typical absolute continuity for classes of dynamically defined measures
Daviaud Edouard Mass transference principle for self-similar measures with overlaps and dimension of self-similar shrinking targets
poster
Neil Manibo Measures arising from Regular Sequences
Will O'Regan Efficiently covering sets with tubes
Chris Williams Multivariate Orthonormal Polynomials on Self-Affine Fractals

Invited Speakers: Titles and Abstracts

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

Abstract: 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 parameters.
Time permitting I also 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.
slides

• Balázs Bárány
Recurrence rates for shifts of finite type

Abstract: In this talk, we study the almost sure recurrence rates of Gibbs measures for Hölder continuous potentials over subshifts of finite type. More precisely, for a given function $$\psi\colon\mathbb{N}\mapsto\mathbb{R}_+$$, let $R_\psi=\{x:d(\sigma^nx,x)\leq\psi(n)\text{ for infinitely many $$n$$'s}\},$ where $$\sigma$$ denotes the left-shift operator. We provide sufficient conditions for $$\mu(R_\psi)=1$$ and sufficient conditions for $$\mu(R_\psi)=0$$. As a corollary of these results, we discover a new critical threshold where the measure of $$R_\psi$$ changes from zero to one related to the Theorem of Iterated Logarithm.
This is a joint work with Demi Allen and Simon Baker.

• Karma Dajani
Random $$\beta$$-transformation on fat Sierpinski gasket

Abstract: We consider the iterated function system (IFS) $f_{\vec{q}}(\vec{z})=\frac{\vec{z}+\vec{q}}{\beta},\,\,\vec{q}\in\{(0,0),(1,0),(0,1)\}.$ As is well known, for $$\beta = 2$$ the attractor, $$S_\beta$$, is a fractal called the Sierpiński gasket (or sieve) and for $$\beta>2$$ it is also a fractal. Our goal is to study greedy, lazy and random $$\beta$$-transformations on the attractor for this IFS with $$1 < \beta < 2$$. For $$1<\beta\leq 3/2$$, $$S_\beta$$ is a triangle and it is shown that the greedy transformation $$T_\beta$$ and the lazy transformation $$L_\beta$$ are isomorphic and they both admit an absolutely continuous invariant measure. We show that all $$\beta$$-expansions of a point $$\vec{z}$$ in $$S_\beta$$ can be generated by a random map $$K_\beta$$ defined on $$\{0,1\}^\mathbb{N}\times\{0,1,2\}^\mathbb{N}\times S_\beta$$ and $$K_\beta$$ has a unique invariant measure of maximal entropy when $$1<\beta\leq\beta_*$$, where $$\beta_*\approx 1.4656$$ is the root of $$x^3-x^2-1=0$$. Furthermore, we show the existence of a $$K_\beta$$-invariant probability measure of the form $$m_1\otimes m_2 \otimes \mu_{\beta}$$, where where $$m_1, m_2$$ are product measures on $$\{0,1\}^\mathbb{N},\{0,1,2\}^\mathbb{N}$$, respectively, and $$\mu_{\beta}$$ is absolutely continuous with respect to the two-dimensional Lebesgue measure $$\lambda_2$$. For $$3/2<\beta\leq \beta^*$$, where $$\beta^*\approx 1.5437$$ is the root of $$x^3-2x^2+2x=2$$, there are radial holes in $$S_\beta$$ and in this case, $$K_\beta$$ is defined on $$\{0,1\}^\mathbb{N}\times S_\beta$$. We also show that it has a unique invariant measure of maximal entropy.
This is a joint work with Wenxia Li and Tingyu Zhang.

• Natalia Jurga
Non-existence of the box dimension for dynamically invariant sets

Abstract: One of the key challenges in the dimension theory of smooth dynamical systems is in establishing whether or not the Hausdorff, lower and upper box dimensions coincide for invariant sets. For sets invariant under conformal dynamics, these three dimensions always coincide. On the other hand, considerable attention has been given to examples of sets invariant under non-conformal dynamics whose Hausdorff and box dimensions do not coincide. These constructions exploit the fact that the Hausdorff and box dimensions quantify size in fundamentally different ways, the former in terms of covers by sets of varying diameters and the latter in terms of covers by sets of fixed diameters. In this talk we introduce the first example of a dynamically invariant set with distinct lower and upper box dimensions. Heuristically, this describes that if size is quantified in terms of covers by sets of equal diameters, a dynamically invariant set can appear bigger when viewed at certain resolutions than at others.

• Antti Käenmäki
Finer geometry of planar self-affine sets.

Abstract: For a planar self-affine set satisfying the strong separation condition, it has been recently proved that under mild assumptions the Hausdorff dimension equals the affinity dimension. In this article, we continue this line of research, and our objective is to acquire more refined geometric information in this setting. In a large class of such non-carpet planar self-affine sets, we characterize Ahlfors regularity, determine the Assouad dimension of the set and its projections, and estimate the Hausdorff dimension of slices. We also demonstrate that the Assouad dimension is not necessarily bounded above by the affinity dimension.
The talk is based on recent work with Balázs Bárány and Han Yu.

• Charlene Kalle
Linear expanding and contracting iterated function systems
In this talk I will introduce a toy model of an iterated function system using piecewise affine expanding and contracting maps and describe the dependence of the behaviour of such systems on the Lyapunov exponent. In particular I will describe the two-point motion: the time evolution of the distance between two typical points under the same choices of maps at each step. This is joint work with Ale Jan Homburg.

• Tom Kempton
Structure in Overlaps of Self-Similar sets and measures.

Given a self-similar iterated function system $$\{T_1,..., T_k\}$$ contracting an interval I on the line, the weak separation condition says (loosely) that there exists a constant $$N$$ such that if two different contracted copies of the interval I have the same length and if they overlap, then they overlap in one of at most $$N$$ ways. In this talk we look beyond the case of the weak separation condition, and explain how it can still be possible to see a lot of structure in the set of overlaps arising from an iterated function system.

• Henna Koivusalo
Impressions on dimensions in iterated function systems

This talk will be an overview of dimension theory in iterated function systems. I will begin by explaining the pressure formulas for self-similar and self-affine sets, and go on to discuss how and when these translate into information about the geometry of the sets. In particular, I will discuss the role of non-overlapping conditions and transversality in dimension calculations. The goal of the talk is to get to a point where I can describe roughly what is the state of the art in dimension theory of self-similar and self-affine sets, or at least some aspects of it. If successful, it should form a coherent whole when combined with the rest of the workshop talks. I might mention some of my own work, apologies to any collaborators.

• István Kolossváry
The $$L^q$$ spectrum of self-affine measures on sponges

In this talk we consider self-affine sponges in $$\mathbb{R}^d$$ 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 $$q\in\mathbb{R}$$. 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.

• Çağrı Sert
Projective fibred fractals

Limit sets of matrix semigroups on projective spaces and associated stationary measures are by-now relatively well-understood under (algebraic) irreducibility assumptions. In a recent work with Aoun, we gave a description of stationary measures on projective spaces under only dynamical assumptions, generalizing previous work in this direction by Benoist--Bru\`{e}re and Aoun--Guivarc'h. The limit sets appearing in these works without the irreducibility assumption (projective fibred fractals) encompass the classical self-affine fractals and can be seen as a generalization thereof. After providing some background, I will talk about our work and focus on projective fibred fractals, making the aforementioned connection with self-affine fractals and hoping to raise some interest in these objects.

• Mike Whittaker
Self-similarity of substitution tiling semigroups

Abstract: Substitution tilings arise from graph iterated function systems. Adding a contraction constant, the attractor recovers the prototiles. On the other hand, without the contraction one obtains an infinite tiling. In this talk I'll introduce substitution tilings and an associated semigroup defined by Kellendonk. I'll show that this semigroup defines a self-similar action on a topological Markov shift that's conjugate to the punctured tiling space. The limit space of the self-similar action turns out to be the Anderson-Putnam complex of the substitution tiling and the inverse limit recovers the translational hull.
This was joint work with Jamie Walton.

• Han Yu
Absolute continuity of projections of Bernoulli measures on carpets and sponges

Abstract: In this talk, we study radial projections of Bernoulli measures on Bedford-McMullen carpets (or sponges). To do this, we introduce the notion of Fourier $$l^1$$ dimension. For Bernoulli measures on carpets, we can compute the Fourier $$l^1$$ dimension with the help of computers (or estimate it with the help of chalks and blackboards). Then we show that for Bernoulli measures with Fourier $$l^1$$ dimension greater than one, their radial projections are absolutely continuous with respect to the Lebesgue measure. We will also introduce other applications of Fourier $$l^1$$ dimension, e.g. the continuity of arithmetic products, rational counting and metric Diophantine approximation on fractals.
Some part of the talk is based on a work in progress with D. Allen, S. Chow and P. Varju.

Lightning Talks and Posters: Titles & Abstracts

• Roope Anttila
(Talk) Assouad dimension of Invariant Measures for Place Dependent Probabilities
I will present a formula for the Assouad dimension of invariant measures for strongly separated conformal iterated function systems with place dependent probabilities.

(Poster) Pointwise Assouad Dimension for Measures
I introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and discuss its properties in relation to the global Assouad dimension. In general, the value of the pointwise Assouad dimension differs from the global counterpart, but in many classical cases, it exhibits an exact dimensionality property, namely it coincides with the global Assouad dimension almost everywhere.

• Amlan Banaji
Intermediate dimensions of Bedford-McMullen carpets

The intermediate dimensions are a family of dimensions which interpolate between Hausdorff and box dimension. With the help of some pictures, we will describe some interesting qualitative features of the graph of the intermediate dimensions of self-affine Bedford-McMullen carpets. This is based on joint work with Istvan Kolossvary.
slides, poster

• George Bender
Intersections of Okamoto's function

A dynamical approach to studying horizontal intersections of a family of fractal curves.

• Daviaud Edouard
Mass transference principle for self-similar measures with overlaps and dimension of self-similar shrinking targets

In this poster/talk, we present new results related to the mass transference principle in the case where the reference measure is not Alfhors regular. We establish a general upper-bound "from ball to arbitrary open set" (as stated in the work of Koivusalo and Rams) which is tractable in the case of self-similar measure without any separation conditions (actually even in the case of weakly conformal C1 IFS's). We use this results to prove that, for any self-similar IFS $$S={f_1 ,...,f_n}$$ of $$\mathbb{R}^d$$ with similarity dimension less than $$d$$ and such that the attractor has dimension equal to its similarity dimension (which is weaker then verifying the open set condition for instance), then, for any $$delta>1$$, the dimension of $$limsup_{i\in {1,...,n}^k ,k \in N}B(f_i (x),|f_i (K) |^\delta)=\dim K / \delta$$. We also give complements of some results established by Baker in arXiv:2104.14249.
poster

• Neil Manibo
Measures arising from Regular Sequences

Regular sequences are generalisations of automatic sequences, where the sequence is now allowed to be unbounded. We will describe how to associate measures to these sequences, and give an overview of their spectral classification (lower local dimension, spectral type, ergodicity) relative to the set of matrices which determines the sequence. This is based on joint work with Michael Coons, James Evans, and Zachary Groth.

• Will O'Regan
Efficiently covering sets with tubes

We introduce the notion of tube-nullity and give some examples of sets with this property.

• Vilma Orgoványi
Orthogonal projections of the random Menger sponge

Using a similar random process to the one which results the fractal percolation sets, starting from the deterministic Menger sponge we get the random Menger sponge. We examine its orthogonal projections from the point of Hausdorff dimension, Lebesgue measure and existence of interior point.

• Rudolf Dániel Prokaj
Continuous Piecewise Linear Iterated Function Systems on the line

Consider the attractor $$\Lambda/) of an iterated function system \(\mathcal{F}$$ on the real line consisting of continuous piecewise linear functions. We assume that the functions of $$\mathcal{F}$$ are strict contractions with non-zero slopes, but they are not necessarily injective. We show that typically the box and Hausdorff dimensions of $$\Lambda$$ are equal.

• Aleksi Pyörälä
Normal numbers in self-conformal sets

During recent years, the prevalence of normal numbers in natural subsets of the reals has been an active research topic in fractal geometry. The general idea is that in the absence of any special arithmetic structure, almost all numbers in a given set should be normal, in every base. In our recent joint work with Balázs Bárány, Antti Käenmäki and Meng Wu we verify this for self-conformal sets on the line.

• Alex Rutar
Geometric and Combinatorial Properties of Self-similar Measures

In this talk, I will discuss a generalization of Feng's matrix product representation of equicontractive self-similar measures in the real line. Then, for measures satisfying the weak separation condition, I will discuss results relating the validity of the multifractal formalism with connectivity properties of a certain finite or infinite graph. This provides new examples of overlapping self-similar measures satisfying the multifractal formalism for all $$q\in\mathbb{R}$$, and also gives insight into the failure of the multifractal formalism for self-similar measures.

Typical absolute continuity for classes of dynamically defined measures

Consider a one-parameter family of iterated function systems on the interval and a family of measures on the corresponding symbolic space, depending on the same parameter. We show that classical transversality results on the typical absolute continuity extend to this setting. Among applications are invariant measures for systems with place dependent probabilities and equilibrium measures for hyperbolic IFSs. This is joint work with Balázs Bárány, Károly Simon and Boris Solomyak.

• Chris Williams
Multivariate Orthonormal Polynomials on Self-Affine Fractals

I will show an IFS structure for orthonormal polynomials over self-affine measures. This structure is also present for all the ingredients needed to form Gaussian quadrature for self-affine measures, and in general, everything needed to make finite elements from fractal shapes. This gives a method for numerical approximation of PDE solutions over certain fractal sets.

• Benthen Zeegers
Critical intermittency in random interval maps

Critical intermittency stands for a type of intermittent dynamics in iterated function systems, caused by an interplay of a superstable fixed point and a repelling fixed point. We consider critical intermittency for IFSs of interval maps and demonstrate the existence of a phase transition when varying probabilities, where the absolutely continuous stationary measure changes between finite and infinite. We discuss further properties of this stationary measure and show that its density is not in $$L^q$$ for any $$q > 1$$. This provides a theory of critical intermittency alongside the theory for the well studied Manneville-Pomeau maps, where the intermittency is caused by a neutral fixed point.
slides

• Xintian Zhang
Hausdorff dimension of eventually always hitting points on self-similar set with SSC

We calculated the Hausdorff dimension of eventually always hitting points on self- similar set with SSC. It is a generalization of Bugeaud and Liao's similar result on b-adic interval map. It is possible to generalize the result further to self-conformal case.