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Combinatorial models for spaces of cubic polynomials

Posted in
Alexander Blokh
U Alabama\MPI
Wed, 2014-06-04 14:15 - 15:15
MPIM Lecture Hall
Parent event: 
Dynamics and Numbers
Parent event: 
Number theory lunch seminar

To construct a model for a connectedness locus of polynomials of degree $d\ge 3$ (cf with Thurston's model of the Mandelbrot set), we define linked geolaminations $\mathcal{L}_1$ and $\mathcal{L}_2$. An accordion is defined as the union of a leaf $\ell$ of $\mathcal{L}_1$ and leaves of $\mathcal{L}_2$ crossing $\ell$. We show that any accordion behaves like a gap of one lamination and prove that the maximal perfect (without isolated leaves) sublaminations of $\mathcal{L}_1$ and $\mathcal{L}_2$ coincide. In the cubic case let $\mathcal{D}_3\subset \mathcal{M}_3$ be the set of all dendritic (with only repelling cycles) polynomials. Let $\mathcal{MD}_3$ be the space of all marked polynomials $(P, c, w)$, where $P\in \mathcal{D}_3$ and $c$, $w$ are critical points of $P$ (perhaps, $c=w$). Let $c^*$ be the co-critical point of $c$ (i.e., $P(c^*)=P(c)$ and, if possible, $c^*\ne c$ ). By Kiwi, to $P\in \mathcal{D}_3$ one associates its lamination $\sim_P$ so that each $x\in J(P)$ corresponds to a convex polygon $G_x$ with vertices in $\mathbb{S}$. We relate to $(P, c, w)\in \mathcal{MD}_3$ its

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