Self-similarity and substitutions of the karyon tilings

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 6, Tome 523 (2023), pp. 83-120

Résumé

Self-similar karyon partitions $\mathcal{T}(\mathbf{m},v)$ with parameters the weight vector $\mathbf{m}$ and the star $v$ are considered. The star $v$ defines the geometry of the parallelepipeds of which the tiling consists of and the weight vector $\mathbf{m}$ sets local rules and periodicity of $\mathcal{T}(\mathbf{m},v)$. A deflation $\bigtriangleup:\mathcal{T}(\mathbf{m},v) \longrightarrow \mathcal{T}^{\bigtriangleup}(\mathbf{m},v)$ is being built, where $\mathcal{T}^{\bigtriangleup}(\mathbf{m},v)=A\mathcal{T}(\mathbf{m},v)$, and $A$ is an affine mapping of the space $\mathbb{R}^{d}$. Deflation replaces the basic polyhedra forming the tiling $\mathcal{T}(\mathbf{m},v)$ by smaller polyhedra. This is the main idea of multidimensional approximations by continued fractions.