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## Combinatorial models for spaces of cubic polynomials

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 $\mathcal{MD}_3$.