Wolfgang Steiner’s research while affiliated with Université Paris Cité and other places

The Knuth Twin Dragon is a compact subset of the plane with fractal boundary of Hausdorff dimension s = (\log \lambda)/(\log \sqrt{2}) , \lambda^{3} = \lambda^{2} + 2 . Although the intersection with a generic line has Hausdorff dimension s-1 , we prove that this does not occur for lines with rational parameters. We further describe the intersection of the Twin Dragon with the two diagonals as well as with various axis parallel lines.

For two real bases q0,q1>1$q_0, q_1 > 1$, we consider expansions of real numbers of the form ∑k=1∞ik/(qi1qi2…qik)$\sum _{k=1}^{\infty } i_k/(q_{i_1}q_{i_2}\ldots q_{i_k})$ with ik∈{0,1}$i_k \in \{0,1\}$, which we call (q0,q1)$(q_0,q_1)$-expansions. A sequence (ik)$(i_k)$ is called a unique (q0,q1)$(q_0,q_1)$-expansion if all other sequences have different values as (q0,q1)$(q_0,q_1)$-expansions, and the set of unique (q0,q1)$(q_0,q_1)$-expansions is denoted by Uq0,q1$U_{q_0,q_1}$. In the special case q0=q1=q$q_0 = q_1 = q$, the set Uq,q$U_{q,q}$ is trivial if q is below the golden ratio and uncountable if q is above the Komornik–Loreti constant. The curve separating pairs of bases (q0,q1)$(q_0, q_1)$ with trivial Uq0,q1$U_{q_0,q_1}$ from those with non-trivial Uq0,q1$U_{q_0,q_1}$ is the graph of a function G(q0)$\mathcal {G}(q_0)$ that we call generalized golden ratio. Similarly, the curve separating pairs (q0,q1)$(q_0, q_1)$ with countable Uq0,q1$U_{q_0,q_1}$ from those with uncountable Uq0,q1$U_{q_0,q_1}$ is the graph of a function K(q0)$\mathcal {K}(q_0)$ that we call generalized Komornik–Loreti constant. We show that the two curves are symmetric in q0$q_0$ and q1$q_1$, that G$\mathcal {G}$ and K$\mathcal {K}$ are continuous, strictly decreasing, hence almost everywhere differentiable on (1,∞)$(1,\infty )$, and that the Hausdorff dimension of the set of q0$q_0$ satisfying G(q0)=K(q0)$\mathcal {G}(q_0)=\mathcal {K}(q_0)$ is zero. We give formulas for G(q0)$\mathcal {G}(q_0)$ and K(q0)$\mathcal {K}(q_0)$ for all q0>1$q_0 > 1$, using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of S-adic sequences including Sturmian and the Thue–Morse sequences are simpler than those of Labarca and Moreira (Ann Inst Henri Poincaré Anal Non Linéaire 23, 683–694, 2006) and Glendinning and Sidorov (Ergod Theory Dyn Syst 35, 1208–1228, 2015), and are relevant also for other open dynamical systems.

The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a long-standing open problem, is then equivalent to injectivity on Christoffel words. A q-analog of these matrix products has been proposed recently by Leclere and Morier-Genoud, and we prove that injectivity on Christoffel words holds for this q-analog. The proof is based on the evaluation at $q = \exp(2\pi i/6)$. Other roots of unity provide some information on the original problem, which corresponds to the case q=1. We also extend the problem to arbitrary words and provide a large family of pairs of words where injectivity does not hold.

It has been a long-standing problem to find good symbolic codings for translations on the d -dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy’s approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples include classical algorithms like the Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the 2-dimensional torus having factor complexity 2n + 1 that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all 3-dimensional toral translations having multiscale bounded remainder sets.

A set of words, also called a language, is letter-balanced if the number of occurrences of each letter only depends on the length of the word, up to a constant. Similarly, a language is factor-balanced if the difference of the number of occurrences of any given factor in words of the same length is bounded. The most prominent example of a letter-balanced but not factor-balanced language is given by the Thue-Morse sequence. We establish connections between the two notions, in particular for languages given by substitutions and, more generally, by sequences of substitutions. We show that the two notions essentially coincide when the sequence of substitutions is proper. For the example of Thue-Morse-Sturmian languages, we give a full characterisation of factor-balancedness.

For two real bases $q_0, q_1>1$, set \begin{equation*} \pi_{q_0, q_1}((i_k)) := \sum_{k=1}^{\infty} \frac{i_k}{q_{i_1}q_{i_2}\cdots q_{i_k}}, \quad (i_k)\in\{0,1\}^\infty. \end{equation*} Then $(i_k)$ is called a $(q_0, q_1)$-expansion of a real number x if $\pi_{q_0, q_1}((i_k)) = x$. Define the set $U_{q_0, q_1} := \big\{\mathbf{u} \in \{0,1\}^\infty \,:\, \pi_{q_0, q_1}(\mathbf{u}) \ne \pi_{q_0, q_1}(\mathbf{v}) \ \text{for all} \ \mathbf{v} \ne \mathbf{u}\big\}$. (Replacing the digits $i_k$ by $d_{i_k}$ with $d_1(q_0{-}1) \ne d_0(q_1{-}1)$ does not change this set.) In the special case $q_0 = q_1 = q$, the set of unique expansions $U_{q, q}$ is trivial if q is below the Golden ratio and uncountable if q is above the Komornik--Loreti constant. In this paper, we investigate the generalized golden ratio and the generalized Komornik--Loreti constant for double bases, defined as \begin{align*} \mathcal{G}(q_0) & := \inf\{q_1>1 \,:\, U_{q_0, q_1} \text { is nontrivial}\}, \\ \mathcal{K}(q_0) & := \inf\{q_1>1 \,:\, U_{q_0, q_1} \text{ is uncountable}\}. \end{align*} We show that the functions $\mathcal{G}(q_0)$ and $\mathcal{K}(q_0)$ are continuous, strictly decreasing, and hence almost everywhere differentiable on $(1,\infty)$. Moreover, we explicitly calculate the functions $\mathcal{G}(q_0)$ and $\mathcal{K}(q_0)$ for all $q_0 > 1$.

We consider digit systems $(A,\mathcal{D})$, where $A \in \mathbb{Q}^{n\times n}$ is an expanding matrix and the digit set $\mathcal{D}$ is a suitable subset of $\mathbb{Q}^n$. To such a system, we associate a self-affine set $\mathcal{F} = \mathcal{F}(A,\mathcal{D})$ that lives in a certain representation space $\mathbb{K}$. If A is an integer matrix, then $\mathbb{K} = \mathbb{R}^n$, while in the general rational case $\mathbb{K}$ contains an additional solenoidal factor. We give a criterion for $\mathcal{F}$ to have positive Haar measure, i.e., for being a rational self-affine tile. We study topological properties of $\mathcal{F}$ and prove some tiling theorems. Our setting is very general in the sense that we allow $(A,\mathcal{D})$ to be a nonstandard digit system. A standard digit system $(A,\mathcal{D})$ is one in which we require $\mathcal{D}$ to be a complete system of residue class representatives w.r.t. a certain naturally chosen residue class ring. Our tools comprise the Frobenius normal form and character theory of locally compact abelian groups.