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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 \({\mathcal{M}}\).

MSC:

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

Software:

GitHub; schottky
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References:

[1] Bandt, C., On the Mandelbrot set for pairs of linear maps, Nonlinearity, 15, 1127-1147, (2002) · Zbl 1008.37027
[2] Barnsley, M. and Harrington, A.. A Mandelbrot set for pairs of linear maps. Phys. D.15 (1985), 421-432. doi:10.1016/S0167-2789(85)80008-7 · Zbl 0582.58023
[3] Bousch, T.. Paires de similitudes. Preprint, 1988; available from the author’s webpage http://topo.math.u-psud.fr/∼bousch/preprints/.
[4] Bousch, T.. Connexité locale et par chemins hölderiens pour les systèmes itérés de fonctions. Preprint, 1992; available from the author’s webpage http://topo.math.u-psud.fr/∼bousch/preprints/.
[5] Calegari, D. and Walker, A.. schottky, software available from https://github.com/dannycalegari/schottky.
[6] Climenhaga, V. and Pesin, Y.. Lectures on Fractal Geometry and Dynamical Systems. American Mathematical Society, Providence, RI, 2009. · Zbl 1186.37003
[7] Epstein, D., Cannon, J., Holt, D., Levy, S., Paterson, M. and Thurston, W.. Word Processing in Groups. Jones and Bartlett, Burlington, MA, 1992. · Zbl 0764.20017
[8] Indlekofer, K., Járai, A. and Kátai, I.. On some properties of attractors generated by iterated function systems. Acta Sci. Math. (Szeged)60 (1995), 411-427. · Zbl 0843.28004
[9] Mercat, P., Semi-groupes fortement automatiques, Bull. Soc. Math. France, 141, 423-479, (2013) · Zbl 1347.20062
[10] Odlyzko, A. and Poonen, B.. Zeros of polynomials with 0, 1 coefficients. Enseign. Math.39 (1993), 317-348. · Zbl 0814.30006
[11] Shmerkin, P. and Solomyak, B.. Zeros of {-1, 0, 1} Power series and connectedness loci for self-affine sets. Exp. Math.15 (2006), 499-511. doi:10.1080/10586458.2006.10128977 · Zbl 1122.30002
[12] Solomyak, B., Mandelbrot set for a pair of line maps: the local geometry, Anal. Theory Appl., 20, 149-157, (2004) · Zbl 1079.28006
[13] Solomyak, B., On the ‘Mandelbrot set’ for pairs of linear maps: asymptotic self-similarity, Nonlinearity, 18, 1927-1943, (2005) · Zbl 1084.37043
[14] Solomyak, B. and Xu, H.. On the ‘Mandelbrot set’ for a pair of linear maps and complex Bernoulli convolutions. Nonlinearity16 (2003), 1733-1749. doi:10.1088/0951-7715/16/5/311 · Zbl 1040.28015
[15] Thurston, W.. Entropy in dimension one. Boundarys in Complex Dynamics: In Celebration of John Milnor’s 80th Birthday. Princeton University Press, Princeton, NJ, 2014, pp. 339-384. · Zbl 1408.37031
[16] Tiozzo, G.. Galois conjugates of entropies of real unimodal maps. Preprint, 2013, arXiv:1310.7647.
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