American Institute of Mathematical Sciences

Advanced
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).   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 (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.  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.  doi: 10.1007/s10711-009-9352-7.  Google Scholar

[6]

Robert J. Daverman, "Decompositions of Manifolds,", Pure and Applied Mathematics, 124 (1986).   Google Scholar

[7]

Adrien Douady and John Hubbard, A proof of Thurston's topological characterization of rational functions,, Acta. Math., 171 (1993), 263.  doi: 10.1007/BF02392534.  Google Scholar

[8]

Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics,, Astérisque, 325 (2009).   Google Scholar

[9]

Peter Haïssinsky and Kevin M. Pilgrim, Finite type coarse expanding conformal dynamics,, Groups Geom. Dyn., 5 (2011), 603.  doi: 10.4171/GGD/141.  Google Scholar

[10]

Volodymyr Nekrashevych, "Self-Similar Groups,", Mathematical Surveys and Monographs, 117 (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, (2000), 349.   Google Scholar

[13]

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

[14]

Mary Rees, A partial description of parameter space of rational maps of degree two. I,, Acta Math., 168 (1992), 11.  doi: 10.1007/BF02392976.  Google Scholar

[15]

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

show all references

References:
[1]

Laurent Bartholdi, Functionally recursive groups,, GAP package, (2011).   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 (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.  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.  doi: 10.1007/s10711-009-9352-7.  Google Scholar

[6]

Robert J. Daverman, "Decompositions of Manifolds,", Pure and Applied Mathematics, 124 (1986).   Google Scholar

[7]

Adrien Douady and John Hubbard, A proof of Thurston's topological characterization of rational functions,, Acta. Math., 171 (1993), 263.  doi: 10.1007/BF02392534.  Google Scholar

[8]

Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics,, Astérisque, 325 (2009).   Google Scholar

[9]

Peter Haïssinsky and Kevin M. Pilgrim, Finite type coarse expanding conformal dynamics,, Groups Geom. Dyn., 5 (2011), 603.  doi: 10.4171/GGD/141.  Google Scholar

[10]

Volodymyr Nekrashevych, "Self-Similar Groups,", Mathematical Surveys and Monographs, 117 (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, (2000), 349.   Google Scholar

[13]

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

[14]

Mary Rees, A partial description of parameter space of rational maps of degree two. I,, Acta Math., 168 (1992), 11.  doi: 10.1007/BF02392976.  Google Scholar

[15]

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

[1]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325
[2]

Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002
[3]

Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409
[4]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390
[5]

Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences