An algebraic characterization of expanding Thurston maps

Previous Article
Weak mixing suspension flows over shifts of finite type are universal
JMD Home
This Issue
Next Article
Ergodic infinite group extensions of geodesic flows on translation surfaces

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
Related Papers

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).
[2]	Martin R. Bridson and André Haefliger, "Metric spaces of non-positive curvature,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319 (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. 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. doi: 10.1007/s10711-009-9352-7.
[6]	Robert J. Daverman, "Decompositions of Manifolds,", Pure and Applied Mathematics, 124 (1986).
[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.
[8]	Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics,, Astérisque, 325 (2009).
[9]	Peter Haïssinsky and Kevin M. Pilgrim, Finite type coarse expanding conformal dynamics,, Groups Geom. Dyn., 5 (2011), 603. doi: 10.4171/GGD/141.
[10]	Volodymyr Nekrashevych, "Self-Similar Groups,", Mathematical Surveys and Monographs, 117 (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, (2000), 349.
[13]	Kevin M. Pilgrim, Julia sets as Gromov boundaries following V. Nekrashevych,, Spring Topology and Dynamical Systems Conference, 29 (2005), 293.
[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.
[15]	Michael Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math., 91 (1969), 175.

show all references

References:

[1]	Laurent Bartholdi, Functionally recursive groups,, GAP package, (2011).
[2]	Martin R. Bridson and André Haefliger, "Metric spaces of non-positive curvature,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319 (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. 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. doi: 10.1007/s10711-009-9352-7.
[6]	Robert J. Daverman, "Decompositions of Manifolds,", Pure and Applied Mathematics, 124 (1986).
[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.
[8]	Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics,, Astérisque, 325 (2009).
[9]	Peter Haïssinsky and Kevin M. Pilgrim, Finite type coarse expanding conformal dynamics,, Groups Geom. Dyn., 5 (2011), 603. doi: 10.4171/GGD/141.
[10]	Volodymyr Nekrashevych, "Self-Similar Groups,", Mathematical Surveys and Monographs, 117 (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, (2000), 349.
[13]	Kevin M. Pilgrim, Julia sets as Gromov boundaries following V. Nekrashevych,, Spring Topology and Dynamical Systems Conference, 29 (2005), 293.
[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.
[15]	Michael Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math., 91 (1969), 175.

[1]	Nikita Selinger. Topological characterization of canonical Thurston obstructions. Journal of Modern Dynamics, 2013, 7 (1) : 99-117. doi: 10.3934/jmd.2013.7.99
[2]	Pedro A. S. Salomão. The Thurston operator for semi-finite combinatorics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 883-896. doi: 10.3934/dcds.2006.16.883
[3]	Volodymyr Nekrashevych. The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$. Journal of Modern Dynamics, 2012, 6 (3) : 327-375. doi: 10.3934/jmd.2012.6.327
[4]	Joshua Du, Liancheng Wang. Dispersion relations for supersonic multiple virtual jets. Conference Publications, 2011, 2011 (Special) : 381-390. doi: 10.3934/proc.2011.2011.381
[5]	Hongming Yang, Dexin Yi, Junhua Zhao, Fengji Luo, Zhaoyang Dong. Distributed optimal dispatch of virtual power plant based on ELM transformation. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1297-1318. doi: 10.3934/jimo.2014.10.1297
[6]	Božidar Jovanović, Vladimir Jovanović. Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5163-5190. doi: 10.3934/dcds.2017224
[7]	Radu Saghin. Note on homology of expanding foliations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 349-360. doi: 10.3934/dcdss.2009.2.349
[8]	Carlangelo Liverani. A footnote on expanding maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3741-3751. doi: 10.3934/dcds.2013.33.3741
[9]	Peter Haïssinsky, Kevin M. Pilgrim. Examples of coarse expanding conformal maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2403-2416. doi: 10.3934/dcds.2012.32.2403
[10]	José F. Alves. Stochastic behavior of asymptotically expanding maps. Conference Publications, 2001, 2001 (Special) : 14-21. doi: 10.3934/proc.2001.2001.14
[11]	Rui Gao, Weixiao Shen. Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2013-2036. doi: 10.3934/dcds.2014.34.2013
[12]	Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403
[13]	Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255
[14]	Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1
[15]	John Erik Fornæss, Brendan Weickert. A quantized henon map. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 723-740. doi: 10.3934/dcds.2000.6.723
[16]	Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927
[17]	Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[18]	Amin Aalaei, Hamid Davoudpour. Two bounds for integrating the virtual dynamic cellular manufacturing problem into supply chain management. Journal of Industrial & Management Optimization, 2016, 12 (3) : 907-930. doi: 10.3934/jimo.2016.12.907
[19]	Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems & Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399
[20]	Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75

2016 Impact Factor: 0.706

American Institute of Mathematical Sciences