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Статья

Combinatorial models for spaces of cubic polynomials

Comptes Rendus Mathematique. 2017. Vol. 355. No. 5. P. 590-595.
Ptacek R., Blokh A., Oversteegen L., Timorin V.

W. Thurston constructed a combinatorial model of the Mandelbrot set M2M2such that there is a continuous and monotone projection of M2M2to this model. We propose the following related model for the space MD3MD3of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)∈MD3(P,c1,c2)∈MD3, then every point z in the Julia set of the polynomial P   defines a unique maximal finite set AzAzof angles on the circle corresponding to the rays, whose impressions form a continuum containing z  . Let G(z)G(z)denote the convex hull of AzAz. The convex sets G(z)G(z)partition the closed unit disk. For (P,c1,c2)∈MD3(P,c1,c2)∈MD3let <img height="16" border="0" style="vertical-align:bottom" width="14" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si6.gif">c1⁎be the co-critical point of  c1c1. We tag the marked dendritic polynomial (P,c1,c2)(P,c1,c2)with the set <img height="18" border="0" style="vertical-align:bottom" width="159" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si14.gif">G(c1⁎)×G(P(c2))⊂D‾×D‾. Tags are pairwise disjoint; denote by <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combtheir collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD3MD3to <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combso that <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combserves as a model for MD3MD3.