Iteration for rational maps in matrix spaces (Q2888639)

Denote with \(\mathcal{ M}(2;\mathbb{C})\) the set of \(2\times 2\) matrices with complex coefficients. In the paper under review, the authors consider several questions concerning the iteration of rational and polynomial self-maps \(\Phi\) of \(\mathcal{ M}(2;\mathbb{C})\). Such a self-map \(\Phi\) is called \textit{compatible with conjugation} if \(A\Phi(M)A^{-1}=\Phi(AMA^{-1})\) for a matrix \(A\), for all \(M\in\mathcal{ M}(2;\mathbb{C})\), whenever this makes sense. It is possible to associate a fibration by \(2\)-planes to the center \(\mathcal{ C}=\{\lambda\mathrm{ Id}: \lambda\in\mathbb{C}\}\) of the ring \(\mathcal{ M}(2;\mathbb{C})\). More precisely, for \(M\in\mathcal{ M}(2;\mathbb{C})\setminus\mathcal{ C}\), let \(\mathcal{ P}(M)\) denote the unique plane containing both \(M\) and \(\mathcal{ C}\): these are the matrices commuting with \(M\), so if \(\Phi\) is compatible with conjugation, then it preserves the fibration by the planes \(\mathcal{ P}(M)\). It is possible to consider the map \(\mathrm{ Inv}:\mathcal{ M}(2;\mathbb{C})\to \mathbb{C}^2\) given by \(\mathrm{ Inv}(M) =(\mathrm{tr}(M), \det(M))\), which takes a matrix \(M\) to its invariants. The authors prove in the present paper that \textsl{if \(\Phi\) is compatible with conjugation, then there is a map \(\mathrm{ Sq}~\Phi:\mathbb{C}^2\to \mathbb{C}^2\) such that \({\mathrm{ Sq}~\Phi\circ \mathrm{ Inv} }= \mathrm{ Inv}\circ \Phi\).} Thus \(\Phi\) is semi-conjugate to a lower-dimensional map, passing indeed to the quotient, modulo the invariant fibration \(\mathcal{ P}\). Moreover, \(\mathrm{ Sq}~\Phi\) contains a big part of the dynamics of \(\Phi\).NEWLINENEWLINEThe authors study in particular the self-map \(\Phi_A\) of \(\mathcal{ M}(2;\mathbb{C})\) given by \(\Phi_A(M)=AM^2\). With such a notation, \(\Phi_{\mathrm{ Id}}\) is the squaring map \(M\mapsto M^2\). The paper contains several results about these maps for different choices of \(A\). Write \(\mathrm{ Aut}(\mathcal{ M}(2;\mathbb{C});{\Phi}_A)\) [resp. \(\mathrm{ Bir}(\mathcal{ M}(2;\mathbb{C});{\Phi}_A)\)] for the set of automorphisms [resp. birational maps] of \(\mathcal{ M}(2;\mathbb{C})\) commuting with \(\Phi_A\). It is clear that for any \(P\in\mathrm{GL}(2;\mathbb{C})\), the map \(\sigma_P(M) = PMP^{-1}\) commutes with \(\Phi_{\mathrm{ Id}}\), as well as the involution by transposition \(M\mapsto M^t\). The authors prove that \textsl{\(\mathrm{ Aut}(\mathcal{ M}(2;\mathbb{C});\Phi_{\mathrm{ Id}})\) is the semi-direct product of the group of maps \(\sigma_P\) with the transposition map}. Moreover, they show that, \textsl{by adding the birational involution \(M\mapsto M^{-1}\), we obtain \(\mathrm{ Bir}(\mathcal{M}(2;\mathbb{C});\Phi_{\mathrm{ Id}})\)}.NEWLINENEWLINEThe authors investigate then veriuous other properties of \(\Phi_A\), particularly in the diagonal case \(A=\mathrm{ Diag}(\lambda,\lambda^{-1})\). They determine, for instance, the closure of the periodic points of \(\Phi_A\) for generic \(\lambda\). Moreovere, they describe the attracting basin of \(\Phi_A\) at the origin. They finally prove that \textsl{if \(\Phi_A\) is not conjugate to \(\Phi_{\mathrm{ Id}}\), then \(\mathrm{ Aut}(\mathcal{ M}(2;\mathbb{C});\Phi_A)\) is isomorphic to a \(\mathbb{C}^*\)-action on \(\mathcal{ M}(2;\mathbb{C})\).}