Coarse expanding conformal dynamics

Haïssinsky, Peter ; Pilgrim, Kevin M.

Astérisque, no. 325 (2009) , 147 p.

@book{AST_2009__325__R1_0,
     author = {Ha{\"\i}ssinsky, Peter and Pilgrim, Kevin M.},
     title = {Coarse expanding conformal dynamics},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {325},
     year = {2009},
     mrnumber = {2662902},
     zbl = {1206.37002},
     language = {en},
     url = {http://www.numdam.org/item/AST_2009__325__R1_0/}
}

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AU  - Pilgrim, Kevin M.
TI  - Coarse expanding conformal dynamics
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PB  - Société mathématique de France
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%0 Book
%A Haïssinsky, Peter
%A Pilgrim, Kevin M.
%T Coarse expanding conformal dynamics
%S Astérisque
%D 2009
%N 325
%I Société mathématique de France
%U http://www.numdam.org/item/AST_2009__325__R1_0/
%G en
%F AST_2009__325__R1_0

Haïssinsky, Peter; Pilgrim, Kevin M. Coarse expanding conformal dynamics. Astérisque, no. 325 (2009), 147 p. http://numdam.org/item/AST_2009__325__R1_0/

Bibliographie
Cité par

[AH94] N. Aoki & K. Hiraide - Topological theory of dynamical systems: recent advances, North-Holland Math. Library, vol. 52, North-Holland Publishing Co., Amsterdam, 1994. | MR | Zbl

[Ahl73] L. V. Ahlfors - Conformal invariants: topics in geometric function theory, Ser. Higher Math., McGraw-Hill Book Co., New York, 1973. | MR | Zbl

[BCM04] A. Blokh, C. Cleveland & M. Misiurewicz - Expanding polymodials, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, p. 253-270. | MR | Zbl

[BD01] J.-Y. Briend & J. Duval - Deux caractérisations de la mesure d'équilibre d'un endomorphisme de $P^{k} (𝐂)$ , Publ. Math. Inst. Hautes Études Sci. 93 (2001), p. 145-159. | MR | Zbl | EuDML | Numdam | DOI

[BDK91] P. Blanchard, R. L. Devaney & L. Keen - The dynamics of complex polynomials and automorphisms of the shift, Invent. Math. 104 (1991), no. 3, p. 545-580. | MR | Zbl | EuDML | DOI

[Ben01] R. L. Benedetto - Reduction, dynamics, and Julia sets of rational functions, J. Number Theory 86 (2001), no. 2, p. 175-195. | MR | Zbl | DOI

[Ber90] V. G. Berkovich - Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr., vol. 33, Amer. Math. Soc., Providence, RI, 1990. | MR | Zbl

[BaHs05] M. H. Baker & L.-C. Hsia - Canonical heights, transfinite diameters, and polynomial dynamics, J. Reine Angew. Math. 585 (2005), p. 61-92. | MR | Zbl | DOI

[BrHa99] M. R. Bridson & A. Haefliger - Metric spaces of non-positive curvature, Grundlehren Math. Wiss., vol. 319, Springer-Verlag, Berlin, 1999. | MR | Zbl

[BHK01] M. Bonk, J. Heinonen & P. Koskela - Uniformizing Gromov hyperbolic spaces, Astérisque, vol. 270, Soc. Math. France, Paris, 2001. | MR | Zbl | Numdam

[BK83] M. Brin & A. Katok - On local entropy, Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, p. 30-38. | MR | Zbl | DOI

[BK02a] M. Bonk & B. Kleiner - Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, p. 127-183. | MR | Zbl | DOI

[BK02b] M. Bonk & B. Kleiner, Rigidity for quasi-Möbius group actions, J. Differential Geom. 61 (2002), no. 1, p. 81-106. | MR | Zbl | DOI

[BK05] M. Bonk & B. Kleiner, Conformal dimension and Gromov hyperbolic groups with $2$ -sphere boundary, Geom. Topol. 9 (2005), p. 219-246. | MR | Zbl | EuDML | DOI

[BM06] M. Bonk & D. Meyer - Topological rational maps and subdivisions, in preparation, 2006.

[Bon06] M. Bonk - Quasiconformal geometry of fractals, International Congress of Mathematicians, vol. II, Eur. Math. Soc., Zürich, 2006, p. 1349-1373. | MR | Zbl

[Bou61] N. Bourbaki - Eléments de mathématique. Première partie. Fascicule II. Livre III: Topologie générale. Chapitre 1 : Structures topologiques. Chapitre 2: Structures uniformes, third éd., Actualités Sci. Indust., vol. 1142, Hermann, Paris, 1961. | Zbl

[Bow98] B. H. Bowditch - A topological characterisation of hyperbolic groups, J. Amer. Math. Soc. 11 (1998), no. 3, p. 643-667. | MR | Zbl | DOI

[Bow99] B. H. Bowditch, Convergence groups and configuration spaces, Geometric group theory down under (Canberra, 1996), de Gruyter, Berlin, 1999, p. 23-54. | MR | Zbl

[BP03] M. Bourdon & H. Pajot - Cohomologie $l_{p}$ et espaces de Besov, J. Reine Angew. Math. 558 (2003), p. 85-108. | MR | Zbl

[BS00] M. Bonk & O. Schramm - Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, p. 266-306. | MR | Zbl | DOI

[Can94] J. W. Cannon - The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), no. 2, p. 155-234. | MR | Zbl | DOI

[CC92] J. W. Cannon & D. Cooper - A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. Amer. Math. Soc. 330 (1992), no. 1, p. 419-431. | MR | Zbl | DOI

[CDP90] M. Coornaert, T. Delzant & A. Papadopoulos - Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov., Lecture Notes in Math., vol. 1441, Springer-Verlag, Berlin, 1990. | MR | Zbl

[CFKP03] J. W. Cannon, W. J. Floyd, R. Kenyon & W. R. Parry - Constructing rational maps from subdivision rules, Conform. Geom. Dyn. 7 (2003), p. 76-102. | MR | Zbl | DOI

[CFP94] J. W. Cannon, W. J. Floyd & W. R. Parry - Squaring rectangles: the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, p. 133-212. | MR | Zbl

[CFP01] J. W. Cannon, W. J. Floyd & W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), p. 153-196. | MR | Zbl | DOI

[CFP06] J. W. Cannon, W. J. Floyd & W. R. Parry, Expansion complexes for finite subdivision rules, I., Conform. Geom. Dyn. 10 (2006), p. 63-99. | MR | Zbl | DOI

[CJY94] L. Carleson, P. W. Jones & J.-C. Yoccoz - Julia and John, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 1, p. 1-30. | MR | Zbl | DOI

[Coo93] M. Coornaert - Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), no. 2, p. 241-270. | MR | Zbl | DOI

[CS98] J. W. Cannon & E. L. Swenson - Recognizing constant curvature discrete groups in dimension $3$ , Trans. Amer. Math. Soc. 350 (1998), no. 2, p. 809-849. | MR | Zbl | DOI

[Dek96] K. Dekimpe - Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Math., vol. 1639, Springer-Verlag, Berlin, 1996. | MR | Zbl

[DH93] A. Douady & J. Hubbard - A proof of Thurston's topological characterization of rational functions, Acta. Math. 171 (1993), no. 2, p. 263-297. | MR | Zbl | DOI

[DL03] K. Dekimpe & K. B. Lee - Expanding maps, Anosov diffeomorphisms and affine structures on infra-nilmanifolds, Topology Appl. 130 (2003), no. 3, p. 259-269. | MR | Zbl | DOI

[DaSa97] G. David & S. Semmes - Fractured fractals and broken dreams, self-similar geometry through metric and measure, Oxford Lecture Ser. Math. Appl., vol. 7, The Clarendon Press Oxford University Press, New York, 1997. | MR | Zbl

[DiSi03] T.-C. Dinh & N. Sibony - Dynamique des applications d'allure polynomiale, J. Math. Pures Appl. (9) 82 (2003), no. 4, p. 367-423. | MR | Zbl | DOI

[Edm76] A. L. Edmonds - Branched coverings and orbit maps, Michigan Math. J. 23 (1976), no. 4, p. 289-301. | MR | Zbl | DOI

A. L. Edmonds - Branched coverings and orbit maps, Michigan Math. J. 23 (1977), no. 4, p. 289-301. | MR | Zbl

[EL89] A. È. Erëmenko & M. Y. Lyubich - The dynamics of analytic transformations, Algebra i Analiz 1 (1989), no. 3, p. 1-70. | MR | Zbl

[Ele97] G. Elek - The $ℓ_{p}$ -cohomology and the conformal dimension of hyperbolic cones, Geom. Dedicata 68 (1997), no. 3, p. 263-279. | MR | Zbl | DOI

[Flo80] W. J. Floyd - Group completions and limit sets of Kleinian groups, Invent Math. 57 (1980), no. 3, p. 205-218. | MR | Zbl | EuDML | DOI

[Fox57] R. H. Fox - Covering spaces with singularities, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, NJ, 1957, p. 243-257. | MR | Zbl

[FRL04] C. Favre & J. Rivera-Letelier - Théorème d'équidistribution de Brolin en dynamique $p$ -adique, C. R. Math. Acad. Sci. Paris 339 (2004), no. 4, p. 271-276. | MR | Zbl | DOI

[FS82] G. B. Folland & E. M. Stein - Hardy spaces on homogeneous groups, Math. Notes, vol. 28, Princeton University Press, Princeton, NJ, 1982. | MR | Zbl

[GdlH90] É. Ghys & P. De La Harpe (eds.) - Sur les groupes hyperboliques d'après Mikhael Gromov, Progr. Math., vol. 83, Birkhäuser Boston Inc., Boston, MA, 1990, papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. | MR | Zbl | DOI

[GM87] F. W. Gehring & G. J. Martin - Discrete quasiconformal groups. I, Proc. London Math. Soc. (3) 55 (1987), no. 2, p. 331-358. | MR | Zbl | DOI

[Gro81] M. Gromov - Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), p. 53-73. | MR | Zbl | Numdam | DOI

[Gro03] M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2) 49 (2003), no. 3-4, p. 217-235. | MR | Zbl

[Hei01] J. Heinonen - Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. | MR | Zbl | DOI

[HMM04] A. Hinkkanen, G. J. Martin & V. Mayer - Local dynamics of uniformly quasiregular mappings, Math. Scand. 95 (2004), no. 1, p. 80-100. | MR | Zbl | DOI

[HP08a] P. Haïssinsky & K. M. Pilgrim - Finite type coarse expanding conformal dynamics, 2008, arXiv: math.DS/0802.1173. | MR | Zbl

[HP08b] P. Haïssinsky & K. M. Pilgrim, Thurston obstructions and ahlfors regular conformal dimension, J. Math. Pures Appl. (9) 90 (2008), no. 5, p. 229-241. | MR | Zbl | DOI

[IM96] T. Iwaniec & G. J. Martin - Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, p. 241-254. | MR | Zbl | EuDML

[KH95] A. Katok & B. Hasselblatt - Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge, 1995, with a supplementary chapter by Katok and L. Mendoza. | MR | Zbl | DOI

[KL05] V. A. Kaimanovich & M. Y. Lyubich - Conformal and harmonic measures on laminations associated with rational maps, Mem. Amer. Math. Soc. 173 (2005), no. 820. | MR | Zbl

[LM97] M. Y. Lyubich & Y. Minsky - Laminations in holomorphic dynamics, J. Differential Geom. 47 (1997), no. 1, p. 17-94. | MR | Zbl | DOI

[Lyu83] M. Y. Lyubich - Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), no. 3, p. 351-385. | MR | Zbl

[Mañ88] R. Mañé - The Hausdorff dimension of invariant probabilities of rational maps, Dynamical systems (Valparaiso 1986), Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, p. 86-117. | MR | Zbl

[Mar97] G. J. Martin - Branch sets of uniformly quasiregular maps, Conform. Geom. Dyn. 1 (1997), p. 24-27. | MR | Zbl | DOI

[Mar04] G. J. Martin, Extending rational maps, Conform. Geom. Dyn. 8 (2004), p. 158-166. | MR | Zbl | DOI

[Mat95] P. Mattila - Geometry of sets and measures in Euclidean spaces, fractals and rectifiability, Cambridge Stud. Adv. Math., vol. 44, Cambridge University Press, Cambridge, 1995. | MR | Zbl

[May97] V. Mayer - Uniformly quasiregular mappings of Lattès type, Conform. Geom. Dyn. 1 (1997), p. 104-111. | MR | Zbl | DOI

[McM94] C.T. Mcmullen - Complex dynamics and renormalization, Princeton University Press, Princeton, NJ, 1994. | MR | Zbl

[McM95] C.T. Mcmullen, The classification of conformal dynamical systems, Current developments in mathematics, Internat. Press, Cambridge, MA, 1995, p. 323-360. | MR | Zbl

[McM96] C.T. Mcmullen, Renormalization and $3$ -manifolds which fiber over the circle, Princeton University Press, Princeton, NJ, 1996. | MR | Zbl | DOI

[McM98a] C.T. Mcmullen, Kleinian groups and John domains, Topology 37 (1998), no. 3, p. 485-496. | MR | Zbl | DOI

[McM98b] C.T. Mcmullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math. 180 (1998), no. 2, p. 247-292. | MR | Zbl | DOI

[McM00] C.T. Mcmullen, Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps, Comment. Math. Helv. 75 (2000), no. 4, p. 535-593. | MR | Zbl | DOI

[Mey02] D. Meyer - Quasisymmetric embedding of self similar surfaces and origami with rational maps, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, p. 461-484. | MR | Zbl | EuDML

[Mil06a] J. Milnor - Dynamics in one complex variable, third ed., Ann. of Math. Stud., vol. 160, Princeton University Press, Princeton, NJ, 2006. | MR | Zbl

[Mil06b] J. Milnor, On Lattès maps, Dynamics on the Riemann sphere, Eur. Math. Soc., Zurich, 2006, p. 9-43. | MR | Zbl

[MM03] G. J. Martin & V. Mayer - Rigidity in holomorphic and quasiregular dynamics, Trans. Amer. Math. Soc. 355 (2003), no. 11, p. 4349-4363. | MR | Zbl | DOI

[MMP06] G. J. Martin, V. Mayer & K. Peltonen - The generalized Lichnerowicz problem: uniformly quasiregular mappings and space forms, Proc. Amer. Math. Soc. 134 (2006), no. 7, p. 2091-2097). | MR | Zbl | DOI

[MP77] M. Misiurewicz & F. Przytycki - Topological entropy and degree of smooth mappings, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 6, p. 573-574. | MR | Zbl

[MS98] C. T. Mcmullen & D. P. Sullivan - Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math. 135 (1998), no. 2, p. 351-395. | MR | Zbl

[MSV99] O. Martio, U. Srebro & J. Väisälä - Normal families, multiplicity and the branch set of quasiregular maps, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 1, p. 231-252. | MR | Zbl | EuDML

[MT88] J. Milnor & W. Thurston - On iterated maps of the interval , Dynamical systems (College Park, MD, 1986-87), Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, p. 465-563. | MR | Zbl

[Nek05] V. Nekrashevych - Self-similar groups, Math. Surveys Monogr., vol. 117, American Mathematical Society, Providence, RI, 2005. | MR | Zbl | DOI

[Pan89a] P. Pansu - Dimension conforme et sphère à l'infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), no. 2, p. 177-212. | MR | Zbl | DOI

[Pan89b] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, p. 1-60. | MR | Zbl | DOI

[Par64] W. Parry - On Rohlin's formula for entropy, Acta Math. Acad. Sci. Hungar. 15 (1964), p. 107-113. | MR | Zbl | DOI

[Pat87] S. J. Patterson - Lectures on measures on limit sets of Kleinian groups, Analytical and geometric aspects of hyperbolic space ((Coventry/Durham, 1984)), London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, p. 281-323. | MR | Zbl

[Pau96] F. Paulin - Un groupe hyperbolique est déterminé par son bord, J. London Math. Soc. (2) 54 (1996), no. 1, p. 50-74. | MR | Zbl | DOI

[Pel99] K. Peltonen - Examples of uniformly quasiregular mappings, Conform. Geom. Dyn. 3 (1999), p. 158-163. | MR | Zbl | DOI

[Pil03] K. M. Pilgrim - Combinations of complex dynamical systems, Lecture Notes in Math., vol. 1827, Springer-Verlag, Berlin, 2003. | MR | Zbl

[PU] F. Przytycki & M. Urbanski - Fractals in the plane - the ergodic theory methods, preprint available at http://www.math.unt.edu/~urbanski/bookl.html. | Zbl

[Ric93] S. Rickman - Quasiregular mappings, Ergeb. Math. Grenzgeb. (3), vol. 26, Springer-Verlag, Berlin, 1993. | MR | Zbl

[RL03a] J. Rivera-Letelier - Dynamique des fonctions rationnelles sur des corps locaux, Geometric methods in dynamics. II, Astérisque, vol. 287, Soc. Math. France, Paris, 2003, p. 147-230. | MR | Zbl | Numdam

[RL03b] J. Rivera-Letelier, Espace hyperbolique $p$ -adique et dynamique des fonctions rationnelles, Compositio Math. 138 (2003), no. 2, p. 199-231. | MR | Zbl | DOI

[Roh49] V. A. Rohlin - On the fundamental ideas of measure theory, Mat. Sbornik N.S. 25(67) (1949), p. 107-150. | MR

[SL00] M. Shishikura & T. Lei - An alternative proof of Mañé's theorem on non-expanding Julia sets, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, p. 265-279. | MR | Zbl

[ST80] J. O. Strömberg & A. Torchinsky - Weights, sharp maximal functions and Hardy spaces, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 3, p. 1053-1056. | MR | Zbl | DOI

[Sto56] S. Stoïlow - Leçons sur les principes topologiques de la théorie des fonctions analytiques. Deuxième édition, augmentée de notes sur les fonctions analytiques et leurs surfaces de Riemann, Gauthier-Villars, Paris, 1956. | MR | Zbl

[SU00] B. O. Stratmann & M. Urbański - The geometry of conformal measures for parabolic rational maps, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 1, p. 141-156. | MR | Zbl | DOI

[SU02] B. O. Stratmann & M. Urbański, Jarník and Julia: a Diophantine analysis for parabolic rational maps for geometrically finite Kleinian groups with parabolic elements, Math. Scand. 91 (2002), no. 1, p. 27-54. | MR | Zbl | DOI

[Sul81] D. Sullivan - On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton NJ, 1981, p. 465-496. | Zbl | MR

[Sul82] D. Sullivan, Seminar on hyperbolic geometry and conformal dynamical systems, preprint, 1982.

[Sul83] D. Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) (Berlin), Lecture Notes in Math., vol. 1007, Springer, 1983, p. 725-752. | MR | Zbl

[Sul85] D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution to the fatou-julia problem on wandering domains, Annals of Mathematics 122 (1985), p. 401-418. | MR | Zbl | DOI

[Tuk81] P. Tukia - A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Math. 6 (1981), no. 1, p. 149-160. | MR | Zbl | DOI

[Tuk86] P. Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), p. 318-346. | MR | Zbl | DOI

[Tuk94] P. Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994), no. 2, p. 157-187. | Zbl | MR

[Tys01] J. T. Tyson - Metric and geometric quasiconformality in Ahlfors regular Loewner spaces, Conform. Geom. Dyn. 5 (2001), p. 21-73. | MR | Zbl | DOI

[Väi71] J. Väisälä - Lectures on $n$ -dimensional quasiconformal mappings, Lecture Notes in Math., vol. 229, Springer-Verlag, Berlin, 1971. | MR | Zbl

[Wal82] P. Walters - An introduction to ergodic theory, Grad. Texts in Math., vol. 79, Springer-Verlag, New York, 1982. | MR | Zbl | DOI

[Zal98] L. Zalcman - Normal families: new perspectives, Bull. Amer. Math. Soc. (N.S.) 35 (1998), no. 3, p. 215-230. | MR | Zbl | DOI

[Zdu90] A. Zdunik - Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, p. 627-649. | MR | Zbl | EuDML | DOI