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Lattès maps and combinatorial expansion

Published online by Cambridge University Press:  11 February 2015

QIAN YIN
QIAN YIN*
Affiliation:
The University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA email [email protected]
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Abstract

A Lattès map $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ is a rational map that is obtained from a finite quotient of a conformal torus endomorphism. We characterize Lattès maps by their combinatorial expansion behavior.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 4 , June 2016 , pp. 1307 - 1342
Copyright
© Cambridge University Press, 2015 

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References

Bonk, M. and Kleiner, B.. Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150 (2002), 127–183.Google Scholar
Bonk, M. and Meyer, D.. Expanding Thurston maps. Preprint, 2010, arXiv:1009.3647v1.Google Scholar
Diestel, R.. Graph Theory (Graduate Texts in Mathematics, 173), 3rd edn. Springer, Berlin, 2005.Google Scholar
Douady, A. and Hubbard, J. H.. A proof of Thurston’s topological characterization of rational functions. Acta Math. 171 (1993), 263–297.Google Scholar
Fatou, P.. Sur les substitutions rationnelles. C. R. Acad. Sci. Paris 164 (1917), 806–808.Google Scholar
Hamenstädt, U.. Entropy-rigidity of locally symmetric spaces of negative curvature. Ann. of Math. (2) 131 (1990), 35–51.Google Scholar
Heinonen, J. and Koskela, P.. Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998), 1–61.Google Scholar
Haïssinsky, P. and Pilgrim, K. M.. Coarse expanding conformal dynamics. Astérisque 325 (2009), 139 pp.Google Scholar
Julia, G.. Mémoire sur l’iteration des fonctions rationnelles. J. Math. Pures Appl. 8 (1918), 47–245.Google Scholar
Lattès, S.. Sur l’iteration des substitutions rationnelles et les fonctions de Poincaré. C. R. Acad. Sci. Paris 166 (1918), 26–28.Google Scholar
Lax, P.. Linear Algebra and its Applications (Pure and Applied Mathematics, 10), 2nd edn. Wiley, New York, 2008.Google Scholar
Mandelbrot, B.. Les Objets Fractals. Flammarion, Paris, 1975.Google Scholar
Mandelbrot, B.. The Fractal Geometry of Nature. W. H. Freeman, San Francisco, CA, 1982.Google Scholar
Meyer, D.. Dimension of elliptic harmonic measure of snowspheres. Illinois J. Math. 53 (2009), 691–721.CrossRefGoogle Scholar
Milnor, J.. On Lattès maps. Dynamics on the Riemann Sphere. European Mathematical Society, Zürich, 2006, pp. 9–43.Google Scholar
Mañé, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4) 16(2) (1983), 193–217.CrossRefGoogle Scholar
Newman, M. H. A.. Elements of the Topology of Plane Sets of Points, 2nd edn. Dover, New York, 1992.Google Scholar
Schröder, E.. Ueber iterirte Functionen. Math. Ann. 3(2) (1871), 296–332.Google Scholar
Sullivan, D.. Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou–Julia problem on wandering domains. Ann. of Math. (2) 122 (1985), 401–418.Google Scholar
Thurston, W.. The geometry and topology of three-manifolds. Lecture Notes, Princeton University, 1980.Google Scholar
Yin, Q.. Thurston maps and asymptotic upper curvature. Preprint, 2011, arXiv:1109.2980.Google Scholar
Zdunik, A.. Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math. 99 (1990), 627–649.Google Scholar