Roots, Schottky semigroups, and a proof of Bandt’s conjecture. (English) Zbl 1393.37062

Summary: In 1985, M. E. Barnsley and A. N. Harrington [Physica D 15, 421–432 (1985; Zbl 0582.58023)] 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 \[ x\mapsto zx\quad \text{and}\quad x\mapsto z(x-1)+1 \] 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 C. Bandt [Nonlinearity 15, No. 4, 1127–1147 (2002; Zbl 1008.37027)] in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique of traps to 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 \({\mathcal{M}}\).


37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
28A80 Fractals
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)


GitHub; schottky
Full Text: DOI arXiv


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