On the dynamic geometry of Kasner polygons with complex parameter (Q6616678)

The paper deals with the dynamics of the sequence of Kasner polygons in the plane. It extends the results by the authors in [Springer Proc. Math. Stat. 418, 213--228 (2023; Zbl 1535.51019)]. Given a polygon with vertices, in complex coordinates, \(A_0^1(a_1)\cdots A_0^k(a_k)\) and a real or complex number \(\alpha\), the vertices of the Kasner iteration are recursively given as follows:\N\[\N\left\{ \begin{aligned} \Na^1_{n+1} &=\alpha a^1_n+(1-\alpha)a^2_n\\\Na^2_{n+1} &=\alpha a^2_n+(1-\alpha)a^3_n\\\N&\cdots\\\Na^{k-1}_{n+1} &=\alpha a^{k-1}_n+(1-\alpha)a^k_n\\\Na^k_{n+1} &=\alpha a^k_n+(1-\alpha)a^1_n\\\N\end{aligned} \right.\N\]\NIn the paper under review the authors investigate geometric features of the dynamics generated by the sequence \(A_n^1\cdots A_n^k\) (or \(a^1_n \cdots a^k_n\)) when \(\alpha\) is complex number.\N\NIn order to present one of the results, define \N\[\NR_j=2\sin\Big(\frac{j}{k}\pi\Big), \quad z_j=\frac{1}{2}-\frac{1}{2}\cot\Big(\frac{j}{k}\pi\Big)i, \quad j=1,\dots,k-1,\N\]\Nand \(D_j=\Big\{z\in \mathbb{C}: |z-z_j|<\frac{1}{R_j} \Big\}\). \N\NThen one has the following.\N\NTheorem. The following assertions hold: the sequence \(A_n^1\cdots A_n^k\), \(n\ge 0\), converges if and only if \(\alpha \in D_1\cap D_{k-1}\) and in this case the limit is the centroid of the initial polygon. If \(\alpha \in \mathrm{int}[(D_1\cap D_{k-1})^c]\), the orbits are divergent.\N\NOther results presented in the paper are about orbits with a finite number of convergent subsequences and the existence of dense orbits.\N\NFor the entire collection see [Zbl 1537.37005].