Cantor bouquets, explosions, and Knaster continua: Dynamics of complex exponentials
This paper is concerned with the topology and dynamics of the complex exponential family \(E_\lambda(z)= \lambda\exp^z\) with a real and positive parameter \(\lambda\). In the Julia set, \(J(E_\lambda)\), of \(E_\lambda\) there are two interesting topological structures: Cantor bouquets and Knaster-like continua. The author describes the construction of each in detail. As the parameter \(\lambda\) passes through the value \(1/e\), the Julia set \(J(E_\lambda)\) undergoes a remarkable transformations. The author shows that \(\bullet\) For \(0\leq\lambda<1/e\), \(J(E_\lambda)\) is a Cantor bouquet, which is a nowhere dense subset of the right half plane. \(\bullet\) At \(\lambda= 1/e\), \(J(E_\lambda)\) undergoes a simple saddle-node bifurcation. At this bifurcation, the attracting fixed point and its entire attracting basin disappear. However, a new topological invariant set arises. \(\bullet\) For \(\lambda>1/e\), \(J(E_\lambda)\) becomes the whole plane. And this new set is a Knaster-like indecomposable continuum on which most orbits cycle toward the orbit of 0 and \(\infty\).
scientific article; zbMATH DE number 1331806