We investigate the question of when a tiling has pure point spectrum, for the class of S-adic tilings, which includes all self-affine tilings. The overlap algorithm by Solomyak is a powerful tool to study this problem for the class of self-affine tilings. We generalize this algorithm for general S-adic tilings, and apply it to a class of block S-adic tilings to show almost all of them have pure point spectra. This is a joint work with Jörg Thuswaldner.

In this talk I will discuss applications of methods of ergodic theory to obtain pointwise asymptotic behaviour for the sum of the digits of some non-regular continued fraction algorithms. The idea is to study the behaviour of trimmed Birkhoff sums for infinite-measure preserving dynamical systems. The talk is based on joint work with Tanja I. Schindler.

We extend the well-known Dumont-Thomas numeration system to $\mathbb{Z}$ by considering two-sided periodic points of a substitution, thus allowing us to represent any integer in $\mathbb{Z}$ by a finite word (starting with 0 when nonnegative and with 1 when negative). We show that an automaton returns the letter at position $n \in \mathbb{Z}$ of the periodic point when fed with the representation of $n$. The numeration system naturally extends to $\mathbb{Z}^d$. We give an equivalent characterization of the numeration system in terms of a total order on a regular language. Lastly, using particular periodic points, we recover the well-known two's complement numeration system and the Fibonacci analogue of the two's complement numeration system.

We study functions related to the classical Brjuno function, namely k-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We then complexify the functional equations which they fulfill and we construct analytic extensions of the k-Brjuno and Wilton functions to the upper half-plane. We study their boundary behaviour using an extension of the continued fraction algorithm to the complex plane. We also prove that the harmonic conjugate of the real k-Brjuno function is continuous at all irrational numbers and has a decreasing jump of π/qk at rational points p/q. This is based on joint work with S. B. Lee, I. Petrykiewicz and T. I. Schindler, the paper is available (open source) at this link: https://link.springer.com/article/10.1007/s00010-023-00967-w

If $\beta$ is an integer, then each $x \in \mathbb{Z}[1/\beta] \cap [0,\infty)$ has finite expansion in base $\beta$. As a generalization of this property for $\beta>1$, the condition (F$_{1}$) that each $x \in \mathbb{N}$ has finite $\beta$-expansion was proposed by Frougny and Solomyak. In this talk, we give a sufficient condition for (F$_{1}$). Moreover we also find $\beta$ with property (F$_{1}$) which does not have positive finiteness property.

Let $b\geq2$ be a positive integer. Then every real number $x\in[0,1]$ admits a $b$-adic representation with digits $a_k$. We call the real $x$ simply normal to base $b$ if every digit $d\in\{0,1,\dots,b-1\}$ occurs with the same frequency in the $b$-ary representation. Furthermore we call $x$ normal to base $b$, if it is simply normal with respect to $b$, $b^2$, $b^3$, etc. Finally we call $x$ absolutely normal if it is normal with respect to all bases $b\geq2$.

In the present talk we want to generalize this notion to normality in measure preserving systems like $\beta$-expansions and continued fraction expansions. Then we show constructions of numbers that are (absolutely) normal with respect to several different expansions.

Ferenczi and Mauduit showed in 1997 that a number represented over an integer base by a Sturmian sequence of digits is transcendental. In this talk we generalise this result to hold for all algebraic number base b of absolute value strictly greater than one. More generally, for a given base b and given irrational number θ, we prove rational linear independence of the set comprising 1 together with all numbers of the above form whose associated digit sequences have slope θ.

We give an application of our main result to the theory of dynamical systems. We show that for a Cantor set C arising as the set of limit points of a contracted rotation f on the unit interval, where f is assumed to have an algebraic slope, all elements of C except its endpoints 0 and 1 are transcendental.

This is joint work with Florian Luca and Joel Ouaknine.

Some years ago M.Lapidus introduced the notion of complex dimensions for a Cantor set in the real line. These occur as poles of the complex Dirichlet series formed from the lengths of the bounded intervals (the “fractal strings”) in the complement of the Cantor set. We will explore further these ideas when the Cantor set is the attractor of an iterated function scheme (concentrating on those whose contractions are a finite set of inverse branches of the usual Gauss map).

A set of shapes is called aperiodic if the shapes admit tilings of the plane, but none that have translational symmetry. A longstanding open problem asks whether a set consisting of a single shape could be aperiodic; such a shape is known as an aperiodic monotile or sometimes an “einstein”. The recently discovered “hat” monotile settles this problem in two dimensions. In this talk I provide necessary background on aperiodicity and related topics in tiling theory, review the history of the search for for an aperiodic monotile, and then discuss the hat and its mathematical properties.

The word complexity function $p(n)$ of a subshift $X$ measures the number of $n$-letter words appearing in sequences in $X$, and $X$ is said to have linear complexity if $p(n)/n$ is bounded. It's been known since work of Ferenczi that subshifts X with linear word complexity function (i.e. $\limsup p(n)/n$ finite) have highly constrained/structured behavior. I'll discuss recent work with Darren Creutz, where we show that if $\limsup p(n)/n < 4/3$, then the subshift $X$ must in fact have measurably discrete spectrum, i.e. it is isomorphic to a compact abelian group rotation. Our proof uses a substitutive/S-adic decomposition for such shifts, and I'll touch on connections to the so-called S-adic Pisot conjecture.

Complex continued fractions (CFs) represent a complex number using a descending fraction with Gaussian integer coefficients. The associated dynamical system is exact (Nakada 1981) with a piecewise-analytic invariant measure (Hensley 2006). Certain higher-dimensional CFs, including CFs over quaternions, octonions, as well as the non-commutative Heisenberg group can be understood in a unified way using the Iwasawa CF framework (L-Vandehey 2022). Under some natural and robust assumptions, ergodicity of the associated systems can then be derived from a connection to hyperbolic geodesic flow, but stronger mixing results and information about the invariant measure remain elusive. Here, we study Iwasawa CFs under a more delicate serendipity assumption that yields the finite range condition, allowing us to extend the Nakada-Hensley results to certain Iwasawa CFs over the quaternions, octonions, and in $\mathbb{R}^3$.

This is joint work with Joseph Vandehey.

Starting from Doeblin's observation on the Poissonian nature of occurrences of large digits in typical continued fraction expansions, I will outline some recent work on rare events in measure preserving systems (including spatiotemporal and local limit theorems) which, in particular, allows us to refine Doeblin's statement in several ways.

(Part of this is joint work with Max Auer.)

Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. In the most classical setting, a $\psi$-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function $\psi$. Khintchine's Theorem provides a beautiful characterisation of the Lebesgue measure of the set of $\psi$-well-approximable numbers and is one of the cornerstone results of Diophantine Approximation. In this talk I will discuss the generalisation of Khintchine's Theorem to the setting of approximation for systems of linear forms. I will focus mainly on the topic of inhomogeneous approximation for systems of linear forms. Time permitting, I may also discuss approximation for systems of linear forms subject to certain primitivity constraints. This talk will be based on joint work with Felipe Ramirez (Wesleyan, US).

Given $\beta \in (1,2]$, let $T$ be the $\beta$-transformation on the unit circle $[0,1)$. For $t \in [0,1)$ let $K(t)$ be the survivor set consisting of all $x$ whose orbit under $T$ never hits the open interval $(0,t)$. Kalle et al. [ETDS, 2020] proved that the Hausdorff dimension function $\dim K(t)$ is a non-increasing Devil's staircase in $t$. So there exists a critical value such that $\dim K(t)$ is vanishing when $t$ is passing through this critical value. In this paper we will describe this critical value and analyze its interesting properties. Our strategy to find the critical value depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems. This is joint work with Pieter Allaart.

We present two families of automatic sequences that define algebraic continued fractions in characteristic $2$. The period-doubling sequence belongs to the first family $\mathcal{P}$; and its sum modulo $2$, the Thue-Morse sequence, belongs to the second family $\mathcal{G}$. The family $\mathcal{G}$ contains all the iterated sums of sequences from the $\mathcal{P}$ and more.

We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers.

This dynamics depends on the Lyapunov exponent.

For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existence of an absolutely continuous stationary measure for the two-point dynamics and discuss its consequences.

For nonnegative Lyapunov exponent and pairs $(M,N)$ that are multiplicatively dependent integers, we provide explicit expressions for absolutely continuous stationary measures of the two-point dynamics. These stationary measures are infinite $\sigma$-finite measures in the case of zero Lyapunov exponent.

This is joint work with Charlene Kalle.

In 2020, Dajani and Kalle investigated invariant measures and frequencies of digits of signed binary expansions arising from a parameterised family of piecewise linear interval maps of constant slope 2. Central to their study was a property called ‘matching,’ where the orbits of the left and right limits of discontinuity points agree after some finite number of steps. We obtain analogous results for a parameterised family of ‘symmetric golden maps’ of constant slope $\beta$, with $\beta$ the golden mean. Matching is again central to our methods, though the dynamics of the symmetric golden maps are more delicate than the binary case. We characterize the matching phenomenon in our setting, present explicit invariant measures and frequencies of digits of signed $\beta$-expansions, and—time permitting—show further implications for a family of piecewise linear maps which arise as jump transformations of the symmetric golden maps.

Joint with Karma Dajani.

Okamoto's functions were introduced in 2005 as a one-parameter family of self-affine functions, which are expressed by ternary expansion of x on the interval [0,1]. By changing the parameter, one can produce interesting examples: Perkins' nowhere differentiable function, Bourbaki-Katsuura function and Cantor's Devil's staircase function.

In this talk, we consider the partial derivative of Okomoto's functions with respect to the parameter a. We place a significant focus on a = 1/3 to describe the properties of a nowhere differentiable function K(x) for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension 1.

This is a joint work with T. Mathis and M.Paizanis (undergraduate students) and N.Dalaklis (graduate student). The talk is very accessible and includes many computer graphics.

In the talk we discuss some previous results on the discrepancy of normal numbers and consider the still open question of Korobov: What is the best possible order of discrepancy $D_N$ in $N$, a sequence $(\{b^n\alpha\})_{n\geq 0}$, $b\geq 2,\in\mathbb{N}$, can have for some real number $\alpha$? If $\lim_{N\to\infty} D_N=0$ then $\alpha$ in called normal in base $b$.

So far the best upper bounds for $D_N$ for explicitly known normal numbers in base $2$ are of the form $ND_N\ll\log^2 N$. The first example is due to Levin (1999), which was later generalized by Becher and Carton (2019). In this talk we discuss the recent result in joint work with Gerhard Larcher that guarantees $ND_N\gg \log^2 N$ for Levin's binary normal number. So EITHER $ND_N\ll \log^2N$ is the best possible order for $D_N$ in $N$ of a normal number OR there exist another example of a binary normal number with a better growth of $ND_N$ in $N$. The recent result for Levin's normal number might support the conjecture that $ND_N\ll \log^2N$ is the best order for $D_N$ in $N$ a normal number can obtain.