Roots, Schottky semigroups, and a proof of Bandt’s conjecture

Abstract

In 1985, Barnsley and Harrington defined a ‘Mandelbrot Set’${\mathcal{M}}$for pairs of similarities: this is the set of complex numbers$z$with$0<|z|<1$for which the limit set of the semigroup generated by the similarities$$\begin{eqnarray}x\mapsto zx\quad \text{and}\quad x\mapsto z(x-1)+1\end{eqnarray}$$is connected. Equivalently,${\mathcal{M}}$is the closure of the set of roots of polynomials with coefficients in$\{-1,0,1\}$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ‘holes’ in${\mathcal{M}}$, and conjectured that these holes were genuine. These holes are very interesting, since they are ‘exotic’ components of the space of (2-generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique oftrapsto construct and certify interior points of${\mathcal{M}}$, and use them to prove Bandt’s conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in${\mathcal{M}}$.