### Number theoretic expansions, algorithms and metrical observations.

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We introduce two-dimensional substitutions generating two-dimensional sequences related to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a ${\mathbb{Z}}^{2}$-action by rotations on the circle. This gives a new geometric interpretation of the Jacobi-Perron algorithm, as a map operating on the parameter space of ${\mathbb{Z}}^{2}$-actions by rotations.

The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type. As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which...

Suppose $A\in G{l}_{d}\left(\mathbb{Z}\right)$ has a 2-dimensional expanding subspace ${E}^{u}$, satisfies a regularity condition, called “good star”, and has ${A}^{*}\ge 0$, where ${A}^{*}$ is an of $A$. A morphism $\theta $ of the free group on $\{1,2,\cdots ,d\}$ is called a of $A$ if it has structure matrix $A$. We show that there is a $\Theta $ whose “boundary substitution” $\theta =\partial \Theta $ is a non-abelianization of $A$. Such a tiling substitution $\Theta $ leads to a self-affine tiling of ${E}^{u}\sim {\mathbb{R}}^{2}$ with ${A}_{u}{:=A|}_{{E}_{u}}\in G{L}_{2}\left(\mathbb{R}\right)$ as its expansion. In the last section we find conditions on $A$ so that ${A}^{*}$ has no negative entries.

Sturmian words are infinite words that have exactly factors of length for every positive integer . A Sturmian word is also defined as a coding over a two-letter alphabet of the orbit of point under the action of the irrational rotation (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give an alternative geometric...

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