Inflation and deflation of the karyon tilings

The substitution transformations of inflation and deflation are defined for the karyon tilings $\mathcal{T}(v)$ of multidimensional tori $\mathbb{T}^d$. Such tilings $\mathcal{T}(v)$ consist of parallelepipeds and are generated by its karyons. Stars $v$, sets of $d+1$ vectors in the space $\mathbb{R}^d$, are frames of the karyons. The interest in karyon tilings is due to their connections with multidimensional continued fractions.