Measures and dimensions in conformal dynamics

Bulletin of the American Mathematical Society

Volumn 40, Issue 3, 2003, Pages 281-321

  Urbański, Mariusz ^a  

^a UNIVERSITY OF NORTH TEXAS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038344485     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-03-00985-6     Document Type: Review

References (113)

1. [ADU] J. Aaronson, M. Denker, M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Transactions of A.M.S. 337 (1993), 495-548. MR 94g:58116
2. [Bea] A.F. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. MR 92j:30026
3. [Bes] A.S. Besicovitch, 'Sets of fractional dimension(IV): On rational approximation to real numbers', Jour. London Math. Soc. 9 (1934), 126-131.
4. [BJ] C. Bishop, P. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), 1-39. MR 98k:22043
5. [BZ] O. Bodart, M. Zinsmeister, Quelques resultats sur la dimension de Hausdorff des ensembles polynomes quadratiques, Fund. Math. 151 (1996), 121-137. MR 97i:30034
6. [Bo1] R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES, 50 (1979), 11-25. MR 81g:57023
7. [Bo2] R. Bowen, Equilibrium states and the ergodic theory for Anosov diffeomorphisms. Lect. Notes in Math. 470, Springer, 1975. MR 56:1364
8. [BS] R. Bowen, C. Series, Markov maps associated with Fuchsian groups, Publ. Math. IHES 50 (1979), 153-179. MR 81b:58026
9. [BK] M. Brin, A. Katok, On local entropy, in Geometric Dynamics, Lect. Notes in Math. 1007 (1983), 30-38, Springer Verlag. MR 85c:58063
10. [By] J. Byrne, Multifractal analysis of parabolic rational maps, Ph.D thesis, Univ. of North Texas (1998).
11. [Ca] L. Carleson, On the support of harmonic measure for sets of Cantor type, Ann. Acad. Sci. Fenn. 10 (1985), 113-123. MR 87b:31002
12. [CG] L. Carleson, T.W. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. MR 94h:30033
13. [CJY] L. Carleson, P. W. Jones, J.-Ch. Yoccoz, Julia and John, Bol. Soc. Bras. Mat. 25 (1994), 1-30. MR 95d:30040
14. [DMNU] M. Denker, D. Mauldin, Z. Nitecki, M. Urbański, Conformal measures for rational functions revisited, Fundamenta Math. 157 (1998), 161-173. MR 99j:58122
15. [DPU] M. Denker, F. Przytycki, M. Urbański, On the transfer operator for rational functions on the Riemann sphere, Ergod. Th. and Dynam. Sys. 16 (1996), 255-266. MR 97e:58197
16. [DR] M. Denker, S. Rohde, On Hausdorff Measures and SBR Measures for Parabolic Rational Maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1763-1769. MR 2001k:37070
17. [DU1] M. Denker, M. Urbański, On the existence of conformal measures, Trans. A.M.S. 328 (1991), 563-587. MR 92k:58155
18. [DU2] M. Denker, M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), 365 - 384. MR 92f:58097
19. [DU3] M. Denker, M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. 43 (1991), 107-118. MR 92k:58153
20. [DU4] M. Denker, M. Urbański, On absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math. 3(1991), 561-579. MR 92k:58154
21. [DU5] M. Denker, M. Urbariski, The capacity of parabolic Julia sets, Math. Zeitsch. 211, (1992), 73-86. MR 93j:30022
22. [DU6] M. Denker, M. Urbański, Geometric measures for parabolic rational maps, Ergod. Th. and Dynam. Sys. 12 (1992), 53-66. MR 93d:58133
23. [DU7] M. Denker, M. Urbański, On Hausdorff measures on Julia sets of subexpanding rational, maps, Israel J. of Math. 76 (1991), 193-214. MR 93g:58078
24. [DU8] M. Denker. M. Urbariski, Ergodic theory of Equilibrium states for rational maps, Nonlinearity 4 (1991), 103-134. MR 92a:58112
25. [DH1] A. Douady, J.H. Hubbard, Etude dynamique des polynomes complexes I,II, Publications mathematique d'Orsay 84-2, 1984; 85-4, 1985. MR 87f:58072a; MR 87f:58072b
26. [DH2] A. Douady, J.H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171.2 (1993), 263-297. MR 94j:58143
27. [DSZ] A. Douady, P. Sentenac, M. Zinsmeister, Implosion parabolique et dimension de Hausdorff, C. R. Acad. Sci, Paris, 325 (Serie 1) (1997), 765-772. MR 98i:58195
28. [Fa] K. Falconer, Fractal geometry, Mathematical Foundations and Applications, John Wiley & Sons, 1990. MR 92j:28008
29. [FLM] A. Freire; A. Lopes; R. Mañé: An invariant measure for rational maps, Bol. Soc. Bras. Mat. 14 (1983), 45-62. MR 85m:58110b
30. [Ge] L. Geyer, Porosity of parabolic Julia sets, Complex Variables Theory Appl. 39 (1999), 191-198. MR 2000e:37055
31. [Go] M.I. Gordin, The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969), 1174-1176. MR 40:5012
32. [GS1] J. Graczyk, S. Smirnov, The Fibonacci Julia set, conformal measures and Hausdorff dimension, Preprint.
33. [GS2] J. Graczyk, S. Smirnov, Collet, Eckmann and Holder, Invent. Math. 133 (1998), 69-96. MR 2000a:37029
34. [Gr] M. Gromov, On the entropy of holomorphic maps, Preprint IHES.
35. [GPS] P. Grzegorczyk, F. Przytycki, W. Szlenk, On iterations of Misiurewicz's rational maps on the Riemann sphere, Ann. Inst. Henri Poincaré, 53 (1990). 431-434. MR 92d:30017
36. n. Lect. Notes in Math. 481, Springer Verlag, 1975. MR 56:15866
37. B. Hasselblatt, A. Katok, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, 1995. MR 96c:58055
38. [Ha] G. Havard, Mesures invariantes pour les fractions rationnelles geometriquement finies, Fund. Math. 160 (1999), 39-61. MR 2000h:37068
39. [Hay] N. Haydn, Convergence of the transfer operator for rational maps, Ergodic Theory Dynam. Systems 19 (1999), 657-669. MR 2000f:37055
40. [HH] D. Heicklen, C. Hoffman, Rational maps are d-adic Bernoulli, Ann. of Math. (2) 156 (2002), 103-114.
41. [HS] S. Heinemann, B. Stratmann, Hausdorff dimension 2 for Julia sets of quadratic polynomials, Math. Z. 237 (2001), 571-583. MR 2002d:37079
42. [HV] R. Hill, S. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Publ. Math. 85 (1997), 193-216. MR 99b:58143
43. [Ja] V. Jarník, Diophantische Approximationen and Hausdorff Mass, Mathematicheskii Sbornik 36 (1929), 371-382.
44. [JM] P. Jones, N. Makarov. Density properties of harmonic measure, Ann. of Math. 142 (1995), 427-455. MR 96k:30027
45. [KS] M. Kesseböhme, B. Stratmann, A multifractal analysis for growth rates and applications to geometrically finite Kleinian groups, Preprint 2001.
46. [KR] P. Koskela, S. Rohde, Hausdorff dimension and mean porosity, Math. Ann. 309 (1997), 593-609. MR 98k:28004
47. [Ly1] M. Lyubich, On a typical behaviour of trajectories for a rational map of sphere, Dokl. Ak. N. U.S.S.R. 268 (1982), 29-32. MR 84f:30036
48. [Ly2] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. Dynam. Sys. 3 (1983), 351-386. MR 85k:58049
49. [Mal] R. Mané, The Hausdorff dimension of invariant probabilities of rational maps, Dynamical Systems, Valparaiso 1986, Lect. Notes in Math. 1331, Springer-Verlag (1988), 86-117. MR 90j:58073
50. [Ma2] R. Mané, On a theorem of Fatou. Bol. Soc. Brasil. Mat. 24 (1993), 1-12. MR 94g:58188
51. [Ma3] R. Mané, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Bras. Mat. 14 (1983), 27-43. MR 85m:58110a
52. [Ma4] R. Mané, On the Bernoulli property of rational maps. Ergod. Th. Dynam. Sys. 5 (1985), 71-88. MR 86i:58082
53. [MM] A. Manning, H. McCluskey, Hausdorff dimension for horseshoes, Ergod. Th. and Dynam. Sys. 3 (1983), 251-260. MR 85j:58127
54. [Man] A. Manning, The dimension of a maximal measure for a polynomial map, Ann. of Math. (2) 119 (1984), 425-430. MR 85i:58068
55. [Mar] M. Martens, The existence of σ-finite invariant measures. Applications to real one-dimensional dynamics, Preprint SUNY Stony Brook IMS preprint 1992/1.
56. [Mat] P. Mattila, Geometry of sets and measures in euclidean spaces, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, 1995. MR 96h:28006
57. [May] V. Mayer, Private communication, 2001.
58. [MC1] C. McMullen, Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps, Comment. Math. Helv. 75 (2000), 535-593. MR 2001m:37089
59. [MC2] C. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math. 180 (1998), 247-292. MR 99f:58172
60. [MC3] C. McMullen. Hausdorff dimension and conformal dynamics III: Computation of dimension, Amer. J. Math. 120 (1998), 691-721. MR 2000d:37055
61. [Mi] M. Misiurewicz, Topological conditional entropy. Studia Math. 55 (1976), 175-200. MR 54:3672
62. [Pal] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. MR 56:8841
63. [Pa2] S. J. Patterson, Lectures on measures on limit sets of Kleinian groups, in Analytical and geometric aspects of hyperbolic space, London Math. Soc., Lecture Notes 111, Cambridge Univ. Press, 1987. MR 99b:58122
64. [Pe] Ya. Pesin, Dimension theory in dynamical systems, University of Chicago Press (1997). MR 99b:58003
65. [PW] Ya. Pesin, H. Weiss, The multifractal analysis of Gibbs measures: Motivation, Mathematical foundation and examples, Chaos 7:1 (1997), 89-106. MR 98e:58130
66. [Pra] E. Prado, Teichmüller distance for some polynomial-like maps, SUNY Stony Brook IMS preprint 1996/2, revision 1997.
67. [Pr1] F. Przytycki, Lyapunov characteristic exponents are non-negative. Proc. Amer. Math. Soc. 119.1 (1993), 309-317. MR 93k:58193
68. [Pr2] F. Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math. 80 (1985), 169-171. MR 86g:30035
69. [Pr3] F. Przytycki, Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Trans. AMS 350.2 (1998), 717-742. MR 98d:58155
70. [Pr4] F. Przytycki, On measure and Hausdorff dimension of Julia sets for holomorphic Collet-Eckmann maps, International conference on dynamical systems, Montevideo 1995, Pitman Research Notes in Math. 362 (1996), 167-181. MR 98i:58198
71. [Pr5] F. Przytycki, Sullivan's classification of conformal expanding repellers, Preprint 1991. to appear in the book "Fractals in the plane - ergodic theory methods" by F. Przytycki and M. Urbański.
72. [Pr6] F. Przytycki. On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Holder continuous functions. Bol. Soc. Bras. Mat. 20 (1990), 95-125. MR 93b:58120
73. [PR] F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets. Fund. Math. 155 (1998), 189-199. MR 2000b:37047
74. [PU1] F. Przytycki, M. Urbański, Fractals in the Plane - the Ergodic Theory Methods, available on the web:http://www.math.unt.edu/urbanski, to appear in Cambridge Univ. Press.
75. [PU2] F. Przytycki, M. Urbański, Rigidity of tame rational functions, Bull. Pol. Acad. Sci., Math., 47.2 (1999), 163-182. MR 2000i:37065
76. [PU3] F. Przytycki, M. Urbański, Porosity of Julia sets of non-recurrent and parabolic Collet-Eckmann rational functions, Ann. Acad. Fenn. 26 (2001), 125-154. MR 2002b:37063
77. [PUZ, I] F. Przytycki, M. Urbański, A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic mapsI, Ann. of Math. 130 (1989), 1-40. MR 911:58115
78. [PUZ, II] F. Przytycki, M. Urbański, A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps II, Studia Math. 97 (1991), 189-225. MR 93d:58140
79. [RT] C. Rogers, S. Taylor, Functions continuous and singular with respect to a Hausdorff measure, Mathematika, 8 (1961), 1-31. MR 24:A200
80. [Ru] D. Ruelle, Thermodynamic formalism, Encyclopedia of Math, and Appl., vol. 5, Addison - Wesley, Reading, Mass., 1978. MR 80g:82017
81. [Sc1] F. Schweiger: Number theoretical endomorphisms with σ-finite invariant measures. Isr. J. Math. 21 (1975), 308-318. MR 52:5608
82. [Sc2] F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Sci. Publ., Oxford University Press, New York, 1995. MR 97h:11083
83. [Sh] M. Shishikura, The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets, Ann. of Math. 147 (1998), 225-267. MR 2000f:37056
84. [Si] C. L. Siegel, Iteration of analytic functions, Ann. of Math. 43 (1942), 607-612. MR 4:76c
85. [Sm] S. Smirnov, Spectral Analysis of Julia sets, Thesis (1996).
86. [St1] B. Stratmann, Fractal dimensions for Jarnik limit sets of geometrically finite Kleinian groups; the semi-classical approach, Ark. for Mat. 33 (1995), 385-403. MR 97a:30056
87. [St2] B. Stratmann, Weak singularity spectra of the Patterson measure for geometrically finite Kleinian groups with parabolic elements, Michigan Math. J. 46 (1999), 573-587. MR 2001a:37059
88. [SU1] B. O. Stratmann, M. Urbański, The Geometry of Conformal Measures for Parabolic Rational Maps, Math. Proc. Cambridge Phil. Soc. 128 (2000), 141-156. MR 2000i:37066
89. [SU2] B. O. Stratmann, M. Urbański, Jarnik and Julia; a Diophantine analysis for parabolic rational maps, Math. Scan. 91 (2002), 27-54.
90. [SU3] B. O. Stratmann, M. Urbański, The box-counting dimension for geometrically finite Kleinian groups, Fundamenta Mathematica 149 (1996), 83-93. MR 96m:30062
91. [SU4] B. O. Stratmann, M. Urbański, Metrical Diophantine analysis for tame parabolic iterated function systems, Preprint 2000.
92. [SV] B. O. Stratmann, S. Velani, The Patterson measure for geometrically finite groups with parabolic elements, new and old, Proc. London Math. Soc. 71 (1995) page 197-220. MR 97f:58023
93. [Su1] D. Sullivan, Seminar on conformal and hyperbolic geometry. Preprint IHES (1982).
94. [Su2] D. Sullivan, Conformal dynamical systems. In: Geometric dynamics, Lect. Notes in Math. 1007, Springer Verlag (1983), 725-752. MR 85m:58112
95. [Su3] D. Sullivan, Quasiconformal homeomorphisms in dynamics, topology, and geometry, Proc. Internat. Congress of Math., Berkeley, Amer. Math. Soc., 1987, 1216-1228. MR 90a:58160
96. [Su4] D. Sullivan, The density at infinity of a discrete group, Inst. Hautes Etudes Sci. Pub. Math. 50 (1979). MR 81b:58031
97. [Su5] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta. Math. 153 (1984), 259-277. MR 86c:58093
98. [Su6] D. Sullivan, Disjoint Spheres, Approximation by Imaginary Quadratic Numbers and the Logarithmic Law for Geodesies, Acta Math. 149 (1982) 215-237. MR 84j:58097
99. [TT] S. J. Taylor, C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. A.M.S. 288 (1985), 679-699. MR 87a:28002
100. [Th] W. Thurston, Three-Dimensional Geometry and Topology, Princeton University Press, 1997. MR 97m:57016
101. [U1] M. Urbański, On Some Aspects of Fractal Dimensions in Higher Dimensional Dynamics, in Proc. of the Göttingen Workshop Problems on Higher Dimensional Complex Dynamics, Mathematica Gottingensis 3 (1995) 18-25.
102. [U2] M. Urbański, On Hausdorff dimension of a Julia set with a rationally indifferent periodic point, Studia Math. 97 (1991), 167-188. MR 93a:58146
103. [U3] M. Urbański, Rational functions with no recurrent critical points, Ergod. Th. and Dynam. Sys. 14 (1994), 391-414. MR 95g:58191
104. [U4] M. Urbański, Geometry and ergodic theory of conformal nonrecurrent dynamics, Ergod. Th. and Dynam. Sys. 17 (1997), 1449-1476. MR 99j:58178
105. [UZ] M. Urbański, A. Zdunik, Hausdorff dimension of harmonic measure for self-conformal sets, Adv. Math. 171 (2002), 1-58.
106. [UZ1] M. Urbański, M. Zinsmeister, Geometry of hyperbolic Julia-Lavaurs sets, Indagationes Math. 12 (2001) 273 - 292.
107. [UZ2] M. Urbański, M. Zinsmeister, Continuity of Hausdorff dimension of Julia-Lavaurs sets as a function of the phase, Journal of Conformal Geometry and Dynamics, 5 (2001), 140-152. MR 2002j:37058
108. [Vo] A. Volberg, On the dimension of harmonic measure of Cantor repellers, Mich. Math. J. 40 (1993), 239-258. MR 95d:30043
109. [Wa1] P. Walters, An introduction to ergodic theory, Springer-Verlag, 1982. MR 84e:28017
110. [Wa2] P. Walters, A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1975), 937 - 971. MR 52:11006
111. [Zd] A. Zdunik, Parabolic orbifolds and the dimension of maximal measure for rational maps, Inv. Math. 99 (1990), 627-649. MR 90m:58120
112. [Zi1] M. Zinsmeister (after A. Douady), Basic parabolic implosions in five days, Preprint 1997/8.
113. [Zi2] M. Zinsmeister, Fleur de Leau-Fatou et dimension de Hausdorff, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 1227-1232. MR 99j:58180