Paul Surer

BOKU University | boku · Institute of Mathematics

We introduce and study mixed triangular labyrinthic fractals, which can be seen as an extension of (generalized) Sierpiński gaskets. This is a new class of fractal dendrites that generalize the self-similar triangular labyrinth fractals studied recently by the authors. A mixed triangular labyrinthic fractal is defined by a triangular labyrinthic pa...

The properties of the intersection of Rauzy fractals associated with substitutions having the same incidence matrix have been studied by several authors. Different techniques have been introduced and used for this purpose, one of them is the balanced pair algorithm. In the present paper we explore the actual limitations of this algorithm. We show t...

We present the zeta-expansion as a complex version of the well-known beta-expansion. It allows us to expand complex numbers with respect to a complex base by using integer digits. Our concepts fits into the framework of the recently published rotational beta-expansions. But we also establish relations with piecewise affine maps of the torus and wit...

In this paper, we associate a primitive substitution with a family of non-integer positional number systems with respect to the same base but with different sets of digits. In this way, we generalize the classical Dumont–Thomas numeration which corresponds to one specific case. Therefore, our concept also covers beta-expansions induced by Parry num...

We define and study a class of fractal dendrites called triangular labyrinth fractals. For the construction, we use triangular labyrinth pattern systems, consisting of two triangular patterns: a white and a yellow one. Correspondingly, we have two fractals: a white and a yellow one. The fractals studied here are self-similar, and fit into the frame...

Coding prescriptions are combinatorial objects linked to a substitution, that is a morphism of the free monoid. Originally they have been introduced in order to code the induced symbolic dynamical system. In the present article we are interested in coding prescriptions of compositions and powers of substitutions. This will provide a very general fr...

In the present paper we are interested in number systems in the ring of integers of cyclotomic number fields in order to obtain a result equivalent to Cobham’s theorem. For this reason we first search for potential bases. This is done in a very general way in terms of canonical number systems. In a second step we analyse pairs of bases in view of t...

Two Rauzy fractals are congruent if they differ by an affine transformation only. We give conditions on unimodular Pisot substitutions in order to ensure the congruence of the Rauzy fractals. We use these results to characterise a large family of substitutions that yield central symmetric Rauzy fractals in terms of the induced language.

Tent maps are continuous composites of two linear functions that act on the unit interval. In the present paper, we describe and analyse a connection between dynamical systems induced by tent maps and the dynamics induced by a certain type of beta-expansion. This relation, which is a weaker form of measure-theoretical conjugacy of dynamical systems...

In the present article, we introduce beta-expansions in the ring $\mathbb{Z}_p$ of $p$-adic integers. We characterise the sets of numbers with eventually periodic and finite expansions.

We develop a theory that allows us to code dynamical systems induced by primitive substitutions continuously as shift of finite type in many different ways. The well-known prefix-suffix coding turns out to correspond to one special case. We precisely analyse the basic properties of these codings (injectivity, coding of the periodic points, properti...

Let $\E$ be a commutative ring with identity and $P\in\E[x]$ be a polynomial. In the present paper we consider digit representations in the residue class ring $\E[x]/(P)$. In particular, we are interested in the question whether each $A\in\E[x]/(P)$ can be represented modulo $P$ in the form $e_0+e_1 X + \cdots + e_h X^h$, where the $e_i\in\E[x]/(P)...

We study aperiodic and periodic tilings induced by the Rauzy fractal and its subtiles associated with beta-substitutions related to the polynomial x3−ax2−bx−1x3−ax2−bx−1 for a≥b≥1a≥b≥1. In particular, we compute the corresponding boundary graphs, describing the adjacencies in the tilings. These graphs are a valuable tool for more advanced studies o...

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intima...

For r ∈ R d define the function τr : Z d → Z d , z = (z 0 , · · · , z d−1) → (z 1 , . . . , z d−1 , − ⌊rz⌋), where rz is the scalar product of the vectors r and z. If each orbit of τr ends up at 0, we call τr a shift radix system. It is a well-known fact that each orbit of τr ends up periodically if the polynomial t d + r d−1 t d−1 + · · · + r 0 as...

Let epsilon is an element of [0, 1), r is an element of R(d) and define the mapping T(r,epsilon). : Z(d) -> Z(d) by T(r,epsilon) (z) = (z(1),...,z(d-1), -[rz + epsilon]) (z = (z(0),...,z(d-1))) If for each Z is an element of Z(d) there is a k is an element of N such that the k-th iterate of Tr,epsilon satisfies T(r)(epsilon)(k) (Z) = 0 we. call T(r...

For r∈Rd define the function τr: Zd → Zd in the following way: τr: Zd → Zd, a=(a1,…,ad)→(a2,…,ad,−⌊ra⌋). τr is called a Shift Radix System (SRS) if ∀a∈Zd ∃k∈N: τrk(a) = 0. In this paper we deal with new results concerning the characterisation of the set D0d:={r∈Rd|τr is an SRS}, especially for d=2. For this purpose we adapt and generalise several r...

Shift radix systems have been introduced by Akiyama et al. as a common generalization of �-expansions and canonical number systems. In the present paper we study a variant of them, so-called sym- metric shift radix systems which were introduced recently by Akiyama and Scheicher. In particular, for d ∈ N and r ∈ Rd let (a = (a1,...,ad)) �r : Zd → Zd...