Dynamics of Word Maps on Groups and Polynomial Maps on Algebras

We introduce the notions of Fatou and Julia sets in the context of word maps on complex Lie groups and polynomial maps on finite-dimensional associative $\mathbb C$-algebras. For the group-theoretic question, we investigate the dynamics of the power map $x \mapsto x^{M}$ on the Lie group $\mathrm{GL}_n(\mathbb C)$, where $M \geq 2$ is an integer. For the algebra-related question, we study polynomial self-maps of $\mathrm{M}_n(\mathbb C)$ induced by monic polynomials in one variable. In both cases, we pin down the explicit description of the Fatou and Julia sets. We also show that there does not exist any wandering Fatou component of the pair $(p,\mathrm M_n(\mathbb C))$ where $p\in\mathbb C[z]$ is a monic polynomial of degree $\geq 2$.