Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Expanding Thurston Maps   We study the dynamics of Thurston maps under iteration. These are branched
covering maps $f$ of 2-spheres $S^2$ with a finite set $\mathop{post}(f)$ of
postcritical points. We also assume that the maps are expanding in a suitable
sense. Every expanding Thurston map $f\: S^2 \to S^2$ gives rise to a type of
fractal geometry on the underlying sphere $S^2$. This geometry is represented
by a class of \emph{visual metrics} $\varrho$ that are associated with the map.
Many dynamical properties of the map are encoded in the geometry of the
corresponding {\em visual sphere}, meaning $S^2$ equipped with a visual metric
$\varrho$. For example, we will see that an expanding Thurston map is
topologically conjugate to a rational map if and only if $(S^2, \varrho)$ is
quasisymmetrically equivalent to the Riemann sphere $\widehat{\mathbf{C}}$. We
also obtain existence and uniqueness results for $f$-invariant Jordan curves
$\mathcal{C}\subset S^2$ containing the set $\mathop{post}(f)$. Furthermore, we
obtain several characterizations of Latt\`{e}s maps.

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The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int