Tame semicascades and cascades generated by affine self-mappings of the $d$-torus

Abstract

We give a complete characterization of the affine self-mappings $\phi$ of the torus ${\mathbb{T}}^d$ that generate Köhler-tame semicascades and cascades. Namely, we show that the semicascade generated by $\phi$ is tame if and only if the matrix $A$ of $\phi$ satisfies $A^p=A^q$, where $p$ and $q$ are some nonnegative integers, $p\ne q$. For cascades the corresponding condition has the form $A^m=I$, where $m$ is some positive integer and $I$ is the identity matrix.

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• Tame semicascades and cascades generated by affine self-mappings of the $d$-torus

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• Tame semicascades and cascades generated by affine self-mappings of the $d$-torus