Expanding Thurston Maps
By M. Bonk, Daniel Meyer. 2010
138 citations recorded by the source corpus.
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 Lattes maps.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a4b707701a14ceac68f7d479e8c22625885b46cf
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In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Uniformization of two-dimensional metric surfaces (2014, methodology): “ or analytic definitions of quasiconformality do not give a good theory in this case, and one has to concentrate on QS maps. See [54] and [27] for the basic properties of QS maps in metric spaces, and [15], [18], [41], [42] for results on QS parametrizations of fractal spaces. In this paper we consid”
- Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups (2010, methodology): “5 of [BM] that with respect to any visual metric as in [BM], the 2-sphere is approximately self-similar.”