Book Review
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MathSciNet review:
2869012
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Book Information:
Authors:
Feliks Przytycki and
Mariusz Urbański
Title:
Conformal fractals: ergodic theory methods
Additional book information:
London Mathematical Society Lecture Note Series, 371, Cambridge University Press,
Cambridge,
2010,
x+354 pp.,
ISBN 978-0-521-43800-1
Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
Alain Connes and Masamichi Takesaki, The flow of weights on factors of type $\textrm {III}$, Tohoku Math. J. (2) 29 (1977), no. 4, 473–575. MR 480760, DOI 10.2748/tmj/1178240493
M. Denker and M. Urbański, Hausdorff measures on Julia sets of subexpanding rational maps, Israel J. Math. 76 (1991), no. 1-2, 193–214. MR 1177340, DOI 10.1007/BF02782852
Alexandre Freire, Artur Lopes, and Ricardo Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 45–62. MR 736568, DOI 10.1007/BF02584744
Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. MR 498471, DOI 10.1007/BF02813304
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
Jane Hawkins, Lebesgue ergodic rational maps in parameter space, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 6, 1423–1447. MR 1992056, DOI 10.1142/S021812740300731X
Gerhard Keller, Equilibrium states in ergodic theory, London Mathematical Society Student Texts, vol. 42, Cambridge University Press, Cambridge, 1998. MR 1618769, DOI 10.1017/CBO9781107359987
Wolfgang Krieger, On ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976), no. 1, 19–70. MR 415341, DOI 10.1007/BF01360278
M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 351–385. MR 741393, DOI 10.1017/S0143385700002030
Ricardo Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 27–43. MR 736567, DOI 10.1007/BF02584743
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation 1952 (1952), no. 71, 55. MR 0047744
C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–423, 623–656. MR 26286, DOI 10.1002/j.1538-7305.1948.tb01338.x
Dennis Sullivan, Quasiconformal homeomorphisms in dynamics, topology, and geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 1216–1228. MR 934326
Terence Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems 28 (2008), no. 2, 657–688. MR 2408398, DOI 10.1017/S0143385708000011
Mariusz Urbański, Measures and dimensions in conformal dynamics, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 281–321. MR 1978566, DOI 10.1090/S0273-0979-03-00985-6
Anna Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, 627–649. MR 1032883, DOI 10.1007/BF01234434
M. Zinmeister, Thermodynamic Formalism and Holomorphic Dynamical Systems, SMF/AMS Texts and Mono., Vol. 2 (2000), AMS; in French (1996), SMF.
References
- A.F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, 132, Springer-Verlag, 1991. MR 1128089 (92j:30026)
- L. Carleson and T.W. Gamelin, Complex Dynamics, Springer-Verlag, 1993. MR 1230383 (94h:30033)
- A. Connes, Classification of injective factors. Cases $II_{1}$, $II_{\infty }$, $III_{\lambda }$, $\lambda \neq 1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 0454659 (56:12908)
- A. Connes and M. Takesaki, The flow of weights on factors of type $\textrm {III}$, Tôhoku Math. J. (2) 29 (1977), no. 4, 473–575. MR 480760 (82a:46069a)
- M. Denker and M. Urbański, Hausdorff measures on Julia sets of subexpanding rational maps. Israel J. Math. 76 (1991), no. 1-2, 193–214. MR 1177340 (93g:58078)
- A. Freire, A. Lopes, and R. Mańé, An invariant measure for rational maps, Bol. Soc. Brasil Mat. 14 (1983), 45–62. MR 736568 (85m:58110b)
- H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Analyse Math. 31 (1977), 204–256. MR 0498471 (58:16583)
- H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. MR 603625 (82j:28010)
- J. Hawkins, Lebesgue ergodic rational maps in parameter space, Internat. J. Bifurcation and Chaos, 6 (2003), Vol. 13, 1423–1447. MR 1992056 (2004e:37065)
- G. Keller, Equilibrium States in Ergodic Theory, London Math. Soc. Student Texts 42, 1998. MR 1618769 (99e:28022)
- W. Krieger, On ergodic flows and the isomorphism of factors. Math. Ann. 223 (1976), no. 1, 1970. MR 0415341 (54:3430)
- M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory and Dynam. Systems 3 (1983), 351–385. MR 741393 (85k:58049)
- R. Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Bras. Math., 14 (1983), no. 1, 27–43. MR 736567 (85m:58110a)
- J. Milnor, Dynamics in One Complex Variable, $3$rd Ed, Princeton Univ. Press, 2006. MR 2193309 (2006g:37070)
- V. A. Rohlin, On the fundamental ideas of measure theory, Trans. Amer. Math. Soc. 71 (1952), 1–54. MR 0047744 (13:924e)
- C.E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–423, 623–656. MR 0026286 (10:133e)
- D. Sullivan, Quasiconformal homeomorphisms in dynamics, topology, and geometry, Proc. Inter. Congress of Math. Berkeley, AMS (1986), 1216–1228. MR 934326 (90a:58160)
- Terence Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory and Dynam. Systems 28 (2008), no. 2, 657688. MR 2408398 (2009k:37012)
- M. Urbański, Measures and dimensions in conformal dynamics. Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 281–321. MR 1978566 (2004f:37063)
- A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), 627–649. MR 1032883 (90m:58120)
- M. Zinmeister, Thermodynamic Formalism and Holomorphic Dynamical Systems, SMF/AMS Texts and Mono., Vol. 2 (2000), AMS; in French (1996), SMF.
Review Information:
Reviewer:
Jane Hawkins
Affiliation:
University of North Carolina, Chapel Hill, North Carolina
Email:
jmh@math.unc.edu
Journal:
Bull. Amer. Math. Soc.
49 (2012), 181-186
DOI:
https://doi.org/10.1090/S0273-0979-2011-01337-4
Published electronically:
May 16, 2011
Review copyright:
© Copyright 2011
American Mathematical Society
