An algebraic characterization of expanding Thurston maps

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October 2012, 6(4): 451-476. doi: 10.3934/jmd.2012.6.451

An algebraic characterization of expanding Thurston maps

Peter Haïssinsky ^1, and Kevin M. Pilgrim ^2,

1.	Université Paul Sabatier, Institut de Mathématiques de Toulouse (IMT), 118 route de Narbonne, 31062 Toulouse Cedex 9, France
2.	Dept. Mathematics, Indiana University, Bloomington, IN 47405

Received May 2012 Published January 2013

Abstract
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Let $f\colon S^2 \to S^2$ be a postcritically finite branched covering map without periodic branch points. We give necessary and sufficient algebraic conditions for $f$ to be homotopic, relative to its postcritical set, to an expanding map $g$.

Keywords: expanding., Thurston map, virtual endomorphism.

Mathematics Subject Classification: Primary: 37F20; Secondary: 37D20, 20E08, 20F65, 54E4.

Citation: Peter Haïssinsky, Kevin M. Pilgrim. An algebraic characterization of expanding Thurston maps. Journal of Modern Dynamics, 2012, 6 (4) : 451-476. doi: 10.3934/jmd.2012.6.451

References:

[1]	Laurent Bartholdi, Functionally recursive groups, GAP package, 2011. Available from: http://www.uni-math.gwdg.de/laurent/FR/.
[2]	Martin R. Bridson and André Haefliger, "Metric spaces of non-positive curvature," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999.
[3]	Mario Bonk and Daniel Meyer, Expanding Thurston maps, arXiv:1009.3647, 2010.
[4]	James W. Cannon, William J. Floyd and Walter R. Parry, Finite subdivision rules, Conform. Geom. Dyn., 5 (2001), 153-196 (electronic). doi: 10.1090/S1088-4173-01-00055-8.
[5]	James W. Cannon, William J. Floyd, Walter R. Parry and Kevin Pilgrim, Subdivision rules and virtual endomorphisms, Geom. Dedicata, 141 (2009), 181-195. doi: 10.1007/s10711-009-9352-7.
[6]	Robert J. Daverman, "Decompositions of Manifolds," Pure and Applied Mathematics, 124, Academic Press, Inc., Orlando, FL, 1986.
[7]	Adrien Douady and John Hubbard, A proof of Thurston's topological characterization of rational functions, Acta. Math., 171 (1993), 263-297. doi: 10.1007/BF02392534.
[8]	Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics, Astérisque, 325 (2009), viii+139 pp. (2010).
[9]	Peter Haïssinsky and Kevin M. Pilgrim, Finite type coarse expanding conformal dynamics, Groups Geom. Dyn., 5 (2011), 603-661. doi: 10.4171/GGD/141.
[10]	Volodymyr Nekrashevych, "Self-Similar Groups," Mathematical Surveys and Monographs, 117, American Mathematical Society, Providence, RI, 2005.
[11]	Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems,, \arXiv{0810.4936}., ().
[12]	Kevin Pilgrim and Tan Lei, Rational maps with disconnected Julia set, Géométrie Complexe et Systèmes Dynamiques (Orsay, 1995), Astérisque, (2000), xiv, 349-384.
[13]	Kevin M. Pilgrim, Julia sets as Gromov boundaries following V. Nekrashevych, Spring Topology and Dynamical Systems Conference, Topology Proc., 29 (2005), 293-316.
[14]	Mary Rees, A partial description of parameter space of rational maps of degree two. I, Acta Math., 168 (1992), 11-87. doi: 10.1007/BF02392976.
[15]	Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.

show all references

References:

[1]	Laurent Bartholdi, Functionally recursive groups, GAP package, 2011. Available from: http://www.uni-math.gwdg.de/laurent/FR/.
[2]	Martin R. Bridson and André Haefliger, "Metric spaces of non-positive curvature," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999.
[3]	Mario Bonk and Daniel Meyer, Expanding Thurston maps, arXiv:1009.3647, 2010.
[4]	James W. Cannon, William J. Floyd and Walter R. Parry, Finite subdivision rules, Conform. Geom. Dyn., 5 (2001), 153-196 (electronic). doi: 10.1090/S1088-4173-01-00055-8.
[5]	James W. Cannon, William J. Floyd, Walter R. Parry and Kevin Pilgrim, Subdivision rules and virtual endomorphisms, Geom. Dedicata, 141 (2009), 181-195. doi: 10.1007/s10711-009-9352-7.
[6]	Robert J. Daverman, "Decompositions of Manifolds," Pure and Applied Mathematics, 124, Academic Press, Inc., Orlando, FL, 1986.
[7]	Adrien Douady and John Hubbard, A proof of Thurston's topological characterization of rational functions, Acta. Math., 171 (1993), 263-297. doi: 10.1007/BF02392534.
[8]	Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics, Astérisque, 325 (2009), viii+139 pp. (2010).
[9]	Peter Haïssinsky and Kevin M. Pilgrim, Finite type coarse expanding conformal dynamics, Groups Geom. Dyn., 5 (2011), 603-661. doi: 10.4171/GGD/141.
[10]	Volodymyr Nekrashevych, "Self-Similar Groups," Mathematical Surveys and Monographs, 117, American Mathematical Society, Providence, RI, 2005.
[11]	Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems,, \arXiv{0810.4936}., ().
[12]	Kevin Pilgrim and Tan Lei, Rational maps with disconnected Julia set, Géométrie Complexe et Systèmes Dynamiques (Orsay, 1995), Astérisque, (2000), xiv, 349-384.
[13]	Kevin M. Pilgrim, Julia sets as Gromov boundaries following V. Nekrashevych, Spring Topology and Dynamical Systems Conference, Topology Proc., 29 (2005), 293-316.
[14]	Mary Rees, A partial description of parameter space of rational maps of degree two. I, Acta Math., 168 (1992), 11-87. doi: 10.1007/BF02392976.
[15]	Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.

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