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Working paper

Combinatorial models for spaces of cubic polynomials

Blokh A., Oversteegen L., Ptacek R., Vladlen Timorin.
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 \emph{linked} geolaminations $\mathcal{L}_1$ and $\mathcal{L}_2$. An \emph{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 \emph{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 \emph{dendritic} (with only repelling cycles) polynomials. Let $\mathcal{MD}_3$ be the space of all \emph{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 \emph{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 \emph{mixed tag} $\mathrm{Tag}(P, c, w)=G_{c^*}\times G_{P(w)}$ and show that mixed tags of distinct marked polynomials from $\mathcal{MD}_3$ are disjoint or coincide. Let $\mathrm{Tag}(\mathcal{MD}_3)^+ = \bigcup_{\mathcal{D}_3}\mathrm{Tag}(P,c,w)$. The sets $\mathrm{Tag}(P, c, w)$ partition $\mathrm{Tag}(\mathcal{MD}_3)^+$ and generate the corresponding quotient space $\mathrm{MT}_3$ of $\mathrm{Tag}(\mathcal{MD}_3)^+$. We prove that $\mathrm{Tag}:\mathcal{MD}_3\to \mathrm{MT}_3$ is continuous so that $\mathrm{MT}_3$ serves as a model space for $\mathcal{MD}_3$.