Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Measures and dimensions in conformal dynamics

Author: Mariusz Urbanski
Journal: Bull. Amer. Math. Soc. 40 (2003), 281-321
MSC (2000): Primary 35F35, 37D35; Secondary 37F15, 37D20, 37D25, 37D45, 37A40, 37A05
Published electronically: April 8, 2003
MathSciNet review: 1978566
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This survey collects basic results concerning fractal and ergodic properties of Julia sets of rational functions of the Riemann sphere. Frequently these results are compared with their counterparts in the theory of Kleinian groups, and this enlarges the famous Sullivan dictionary. The topics concerning Hausdorff and packing measures and dimensions are given most attention. Then, conformal measures are constructed and their relations with Hausdorff and packing measures are discussed throughout the entire article. Also invariant measures absolutely continuous with respect to conformal measures are touched on. While the survey begins with facts concerning all rational functions, much time is devoted toward presenting the well-developed theory of hyperbolic and parabolic maps, and in Section 3 the class NCP is dealt with. This class consists of such rational functions $f$ that all critical points of $f$ which are contained in the Julia set of $f$ are non-recurrent. The NCP class comprises in particular hyperbolic, parabolic and subhyperbolic maps. Our last section collects some recent results about other subclasses of rational functions, e.g. Collet-Eckmann maps and Fibonacci maps. At the end of this article two appendices are included which are only loosely related to Sections 1-4. They contain a short description of tame mappings and the theory of equilibrium states and Perron-Frobenius operators associated with Hölder continuous potentials.

References [Enhancements On Off] (What's this?)

  • [ADU] J. Aaronson, M. Denker, M. Urbanski, Ergodic theory for Markov fibred systems and parabolic rational maps, Transactions of A.M.S. 337 (1993), 495-548. MR 94g:58116
  • [Bea] A.F. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. MR 92j:30026
  • [Bes] A.S. Besicovitch, `Sets of fractional dimension(IV): On rational approximation to real numbers', Jour. London Math. Soc. 9 (1934), 126-131.
  • [BJ] C. Bishop, P. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), 1-39. MR 98k:22043
  • [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
  • [Bo1] R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES, 50 (1979), 11-25. MR 81g:57023
  • [Bo2] R. Bowen, Equilibrium states and the ergodic theory for Anosov diffeomorphisms. Lect. Notes in Math. 470, Springer, 1975. MR 56:1364
  • [BS] R. Bowen, C. Series, Markov maps associated with Fuchsian groups, Publ. Math. IHES 50 (1979), 153-179. MR 81b:58026
  • [BK] M. Brin, A. Katok, On local entropy, in Geometric Dynamics, Lect. Notes in Math. 1007 (1983), 30-38, Springer Verlag. MR 85c:58063
  • [By] J. Byrne, Multifractal analysis of parabolic rational maps, Ph.D thesis, Univ. of North Texas (1998).
  • [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
  • [CG] L. Carleson, T.W. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. MR 94h:30033
  • [CJY] L. Carleson, P. W. Jones, J.-Ch. Yoccoz, Julia and John, Bol. Soc. Bras. Mat. 25 (1994), 1-30. MR 95d:30040
  • [DMNU] M. Denker, D. Mauldin, Z. Nitecki, M. Urbanski, Conformal measures for rational functions revisited, Fundamenta Math. 157 (1998), 161-173. MR 99j:58122
  • [DPU] M. Denker, F. Przytycki, M. Urbanski, On the transfer operator for rational functions on the Riemann sphere, Ergod. Th. and Dynam. Sys. 16 (1996), 255-266. MR 97e:58197
  • [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
  • [DU1] M. Denker, M. Urbanski, On the existence of conformal measures, Trans. A.M.S. 328 (1991), 563-587. MR 92k:58155
  • [DU2] M. Denker, M. Urbanski, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), 365 - 384. MR 92f:58097
  • [DU3] M. Denker, M. Urbanski, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. 43 (1991), 107-118. MR 92k:58153
  • [DU4] M. Denker, M. Urbanski, On absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math. 3(1991), 561-579. MR 92k:58154
  • [DU5] M. Denker, M. Urbanski, The capacity of parabolic Julia sets, Math. Zeitsch. 211, (1992), 73-86. MR 93j:30022
  • [DU6] M. Denker, M. Urbanski, Geometric measures for parabolic rational maps, Ergod. Th. and Dynam. Sys. 12 (1992), 53-66. MR 93d:58133
  • [DU7] M. Denker, M. Urbanski, On Hausdorff measures on Julia sets of subexpanding rational maps, Israel J. of Math. 76 (1991), 193-214. MR 93g:58078
  • [DU8] M. Denker, M. Urbanski, Ergodic theory of Equilibrium states for rational maps, Nonlinearity 4 (1991), 103-134. MR 92a:58112
  • [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
  • [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
  • [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
  • [Fa] K. Falconer, Fractal geometry, Mathematical Foundations and Applications, John Wiley & Sons, 1990. MR 92j:28008
  • [FLM] A. Freire; A. Lopes; R. Mañé: An invariant measure for rational maps, Bol. Soc. Bras. Mat. 14 (1983), 45-62. MR 85m:58110b
  • [Ge] L. Geyer, Porosity of parabolic Julia sets, Complex Variables Theory Appl. 39 (1999), 191-198. MR 2000e:37055
  • [Go] M.I. Gordin, The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969), 1174-1176. MR 40:5012
  • [GS1] J. Graczyk, S. Smirnov, The Fibonacci Julia set, conformal measures and Hausdorff dimension, Preprint.
  • [GS2] J. Graczyk, S. Smirnov, Collet, Eckmann and Hölder, Invent. Math. 133 (1998), 69-96. MR 2000a:37029
  • [Gr] M. Gromov, On the entropy of holomorphic maps, Preprint IHES.
  • [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
  • [Gu] M. de Guzmán, Differentiation of integrals in $\mathbb{R}^{n}$. Lect. Notes in Math. 481, Springer Verlag, 1975. MR 56:15866
  • [HK] 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
  • [Ha] G. Havard, Mesures invariantes pour les fractions rationnelles geometriquement finies, Fund. Math. 160 (1999), 39-61. MR 2000h:37068
  • [Hay] N. Haydn, Convergence of the transfer operator for rational maps, Ergodic Theory Dynam. Systems 19 (1999), 657-669. MR 2000f:37055
  • [HH] D. Heicklen, C. Hoffman, Rational maps are $d$-adic Bernoulli, Ann. of Math. (2) 156 (2002), 103-114.
  • [HS] S. Heinemann, B. Stratmann, Hausdorff dimension 2 for Julia sets of quadratic polynomials, Math. Z. 237 (2001), 571-583. MR 2002d:37079
  • [HV] R. Hill, S. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Publ. Math. 85 (1997), 193-216. MR 99b:58143
  • [Ja] V. Jarník, Diophantische Approximationen and Hausdorff Mass, Mathematicheskii Sbornik 36 (1929), 371-382.
  • [JM] P. Jones, N. Makarov, Density properties of harmonic measure, Ann. of Math. 142 (1995), 427-455. MR 96k:30027
  • [KS] M. Kesseböhme, B. Stratmann, A multifractal analysis for growth rates and applications to geometrically finite Kleinian groups, Preprint 2001.
  • [KR] P. Koskela, S. Rohde, Hausdorff dimension and mean porosity, Math. Ann. 309 (1997), 593-609. MR 98k:28004
  • [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
  • [Ly2] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. Dynam. Sys. 3 (1983), 351-386. MR 85k:58049
  • [Ma1] 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
  • [Ma2] R. Mané, On a theorem of Fatou. Bol. Soc. Brasil. Mat. 24 (1993), 1-12. MR 94g:58188
  • [Ma3] R. Mané, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Bras. Mat. 14 (1983), 27-43. MR 85m:58110a
  • [Ma4] R. Mané, On the Bernoulli property of rational maps. Ergod. Th. Dynam. Sys. 5 (1985), 71-88. MR 86i:58082
  • [MM] A. Manning, H. McCluskey, Hausdorff dimension for horseshoes, Ergod. Th. and Dynam. Sys. 3 (1983), 251-260. MR 85j:58127
  • [Man] A. Manning, The dimension of a maximal measure for a polynomial map, Ann. of Math. (2) 119 (1984), 425-430. MR 85i:58068
  • [Mar] M. Martens, The existence of $\sigma$-finite invariant measures, Applications to real one-dimensional dynamics, Preprint SUNY Stony Brook IMS preprint 1992/1.
  • [Mat] P. Mattila, Geometry of sets and measures in euclidean spaces, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, 1995. MR 96h:28006
  • [May] V. Mayer, Private communication, 2001.
  • [Mc1] C. McMullen, Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps, Comment. Math. Helv. 75 (2000), 535-593. MR 2001m:37089
  • [Mc2] C. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math. 180 (1998), 247-292. MR 99f:58172
  • [Mc3] C. McMullen, Hausdorff dimension and conformal dynamics III: Computation of dimension, Amer. J. Math. 120 (1998), 691-721. MR 2000d:37055
  • [Mi] M. Misiurewicz, Topological conditional entropy. Studia Math. 55 (1976), 175-200. MR 54:3672
  • [Pa1] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. MR 56:8841
  • [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 89b:58122
  • [Pe] Ya. Pesin, Dimension theory in dynamical systems, University of Chicago Press (1997). MR 99b:58003
  • [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
  • [Pra] E. Prado, Teichmüller distance for some polynomial-like maps, SUNY Stony Brook IMS preprint 1996/2, revision 1997.
  • [Pr1] F. Przytycki, Lyapunov characteristic exponents are non-negative. Proc. Amer. Math. Soc. 119.1 (1993), 309-317. MR 93k:58193
  • [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
  • [Pr3] F. Przytycki, Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Trans. AMS 350.2 (1998), 717-742. MR 98d:58155
  • [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
  • [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. Urbanski.
  • [Pr6] F. Przytycki, On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions. Bol. Soc. Bras. Mat. 20 (1990), 95-125. MR 93b:58120
  • [PR] F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets. Fund. Math. 155 (1998), 189-199. MR 2000b:37047
  • [PU1] F. Przytycki, M. Urbanski, Fractals in the Plane - the Ergodic Theory Methods, available on the web:$\tilde$urbanski, to appear in Cambridge Univ. Press.
  • [PU2] F. Przytycki, M. Urbanski, Rigidity of tame rational functions, Bull. Pol. Acad. Sci., Math., 47.2 (1999), 163-182. MR 2000i:37065
  • [PU3] F. Przytycki, M. Urbanski, Porosity of Julia sets of non-recurrent and parabolic Collet-Eckmann rational functions, Ann. Acad. Fenn. 26 (2001), 125-154. MR 2002b:37063
  • [PUZ, I] F. Przytycki, M. Urbanski, A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps I, Ann. of Math. 130 (1989), 1-40. MR 91i:58115
  • [PUZ, II] F. Przytycki, M. Urbanski, A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps II, Studia Math. 97 (1991), 189-225. MR 93d:58140
  • [RT] C. Rogers, S. Taylor, Functions continuous and singular with respect to a Hausdorff measure, Mathematika, 8 (1961), 1-31. MR 24:A200
  • [Ru] D. Ruelle, Thermodynamic formalism, Encyclopedia of Math. and Appl., vol. 5, Addison - Wesley, Reading, Mass., 1978. MR 80g:82017
  • [Sc1] F. Schweiger: Number theoretical endomorphisms with $\sigma $-finite invariant measures. Isr. J. Math. 21 (1975), 308-318. MR 52:5608
  • [Sc2] F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Sci. Publ., Oxford University Press, New York, 1995. MR 97h:11083
  • [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
  • [Si] C. L. Siegel, Iteration of analytic functions, Ann. of Math. 43 (1942), 607-612. MR 4:76c
  • [Sm] S. Smirnov, Spectral Analysis of Julia sets, Thesis (1996).
  • [St1] B. Stratmann, Fractal dimensions for Jarnik limit sets of geometrically finite Kleinian groups; the semi-classical approach, Ark. för Mat. 33 (1995), 385-403. MR 97a:30056
  • [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
  • [SU1] B. O. Stratmann, M. Urbanski, The Geometry of Conformal Measures for Parabolic Rational Maps, Math. Proc. Cambridge Phil. Soc. 128 (2000), 141-156. MR 2000i:37066
  • [SU2] B. O. Stratmann, M. Urbanski, Jarnik and Julia; a Diophantine analysis for parabolic rational maps, Math. Scan. 91 (2002), 27-54.
  • [SU3] B. O. Stratmann, M. Urbanski, The box-counting dimension for geometrically finite Kleinian groups, Fundamenta Mathematica 149 (1996), 83-93. MR 96m:30062
  • [SU4] B. O. Stratmann, M. Urbanski, Metrical Diophantine analysis for tame parabolic iterated function systems, Preprint 2000.
  • [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
  • [Su1] D. Sullivan, Seminar on conformal and hyperbolic geometry. Preprint IHES (1982).
  • [Su2] D. Sullivan, Conformal dynamical systems. In: Geometric dynamics, Lect. Notes in Math. 1007, Springer Verlag (1983), 725-752. MR 85m:58112
  • [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
  • [Su4] D. Sullivan, The density at infinity of a discrete group, Inst. Hautes Etudes Sci. Pub. Math. 50 (1979). MR 81b:58031
  • [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
  • [Su6] D. Sullivan, Disjoint Spheres, Approximation by Imaginary Quadratic Numbers and the Logarithmic Law for Geodesics, Acta Math. 149 (1982) 215-237. MR 84j:58097
  • [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
  • [Th] W. Thurston, Three-Dimensional Geometry and Topology, Princeton University Press, 1997. MR 97m:57016
  • [U1] M. Urbanski, 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.
  • [U2] M. Urbanski, On Hausdorff dimension of a Julia set with a rationally indifferent periodic point, Studia Math. 97 (1991), 167-188. MR 93a:58146
  • [U3] M. Urbanski, Rational functions with no recurrent critical points, Ergod. Th. and Dynam. Sys. 14 (1994), 391-414. MR 95g:58191
  • [U4] M. Urbanski, Geometry and ergodic theory of conformal nonrecurrent dynamics, Ergod. Th. and Dynam. Sys. 17 (1997), 1449-1476. MR 99j:58178
  • [UZ] M. Urbanski, A. Zdunik, Hausdorff dimension of harmonic measure for self-conformal sets, Adv. Math. 171 (2002), 1-58.
  • [UZ1] M. Urbanski, M. Zinsmeister, Geometry of hyperbolic Julia-Lavaurs sets, Indagationes Math. 12 (2001) 273 - 292.
  • [UZ2] M. Urbanski, 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
  • [Vo] A. Volberg, On the dimension of harmonic measure of Cantor repellers, Mich. Math. J. 40 (1993), 239-258. MR 95d:30043
  • [Wa1] P. Walters, An introduction to ergodic theory, Springer-Verlag, 1982. MR 84e:28017
  • [Wa2] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math. 97 (1975), 937 - 971. MR 52:11006
  • [Zd] A. Zdunik, Parabolic orbifolds and the dimension of maximal measure for rational maps, Inv. Math. 99 (1990), 627-649. MR 90m:58120
  • [Zi1] M. Zinsmeister (after A. Douady), Basic parabolic implosions in five days, Preprint 1997/8.
  • [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

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 35F35, 37D35, 37F15, 37D20, 37D25, 37D45, 37A40, 37A05

Retrieve articles in all journals with MSC (2000): 35F35, 37D35, 37F15, 37D20, 37D25, 37D45, 37A40, 37A05

Additional Information

Mariusz Urbanski
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430

Received by editor(s): December 22, 1999
Received by editor(s) in revised form: January 8, 2003
Published electronically: April 8, 2003
Additional Notes: Research partially supported by NSF Grant DMS 9801583
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society