A holomorphic correspondence at the boundary of the Klein combination locus (Q1950986)

For values \(a\) which are in the Klein combination locus \(\mathcal{K}\) and which are not in the connectivity locus, the authors study the behaviour of holomorphic correspondences \(\mathcal{F}_{a} \) as \(a\) approaches the outer boundary of \(\mathcal{K}\). Moreover, they investigate what happens when \(a\) actually reaches that boundary. In their analysis they focus on a certain particular boundary point of \(\mathcal{K}\), which they call Penrose point, and investigate the dynamical behaviour of the associated correspondence which has this Penrose point as its parameter value. In particular, they obtain that at this particular point the complement of the corresponding limit set has four components and that for each of these components the subcorrespondence, which stabilises it, is conjugate to a Fuchsian group. In this way they reveal an explicit holomorphic correspondence on the Riemann sphere of very interesting dynamical and fractal nature.