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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

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/. Google Scholar

[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.  Google Scholar

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Mario Bonk and Daniel Meyer, Expanding Thurston maps, arXiv:1009.3647, 2010. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[6]

Robert J. Daverman, "Decompositions of Manifolds," Pure and Applied Mathematics, 124, Academic Press, Inc., Orlando, FL, 1986.  Google Scholar

[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.  Google Scholar

[8]

Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics, Astérisque, 325 (2009), viii+139 pp. (2010).  Google Scholar

[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.  Google Scholar

[10]

Volodymyr Nekrashevych, "Self-Similar Groups," Mathematical Surveys and Monographs, 117, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[11]

Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems,, \arXiv{0810.4936}., ().   Google Scholar

[12]

Kevin Pilgrim and Tan Lei, Rational maps with disconnected Julia set, Géométrie Complexe et Systèmes Dynamiques (Orsay, 1995), Astérisque, (2000), xiv, 349-384.  Google Scholar

[13]

Kevin M. Pilgrim, Julia sets as Gromov boundaries following V. Nekrashevych, Spring Topology and Dynamical Systems Conference, Topology Proc., 29 (2005), 293-316.  Google Scholar

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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.  Google Scholar

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Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.  Google Scholar

show all references

References:
[1]

Laurent Bartholdi, Functionally recursive groups, GAP package, 2011. Available from: http://www.uni-math.gwdg.de/laurent/FR/. Google Scholar

[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.  Google Scholar

[3]

Mario Bonk and Daniel Meyer, Expanding Thurston maps, arXiv:1009.3647, 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Robert J. Daverman, "Decompositions of Manifolds," Pure and Applied Mathematics, 124, Academic Press, Inc., Orlando, FL, 1986.  Google Scholar

[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.  Google Scholar

[8]

Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics, Astérisque, 325 (2009), viii+139 pp. (2010).  Google Scholar

[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.  Google Scholar

[10]

Volodymyr Nekrashevych, "Self-Similar Groups," Mathematical Surveys and Monographs, 117, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[11]

Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems,, \arXiv{0810.4936}., ().   Google Scholar

[12]

Kevin Pilgrim and Tan Lei, Rational maps with disconnected Julia set, Géométrie Complexe et Systèmes Dynamiques (Orsay, 1995), Astérisque, (2000), xiv, 349-384.  Google Scholar

[13]

Kevin M. Pilgrim, Julia sets as Gromov boundaries following V. Nekrashevych, Spring Topology and Dynamical Systems Conference, Topology Proc., 29 (2005), 293-316.  Google Scholar

[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.  Google Scholar

[15]

Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.  Google Scholar

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