Rational families converging to a family of exponential maps
Summary: We analyze the dynamics of a sequence of families of non-polynomial rational maps, \(\{f_{a,d}\}\), for \(a \in \mathbb C^*= \mathbb C \setminus \{0\}, d \geq 2\). For each \(d\), \(\{f_{a,d}\}\) is a family of rational maps of degree \(d\) of the Riemann sphere parametrized by \(a \in \mathbb C^*\). For each \(a \in \mathbb C^*\), as \(d \rightarrow \infty\), \(f_{a,d}\) converges uniformly on compact sets to a map \(f_a\) that is conformally conjugate to a transcendental entire map on \(\mathbb C\). We study how properties of the families \(f_{a,d}\) contribute to our understanding of the dynamical properties of the limiting family of maps. We show all families have a common connectivity locus; moreover the rational maps contain some well-studied examples.
scientific article; zbMATH DE number 7036249
scientific article