Book Review
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MathSciNet review:
2731657
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Book Information:
Authors:
A. A. Pinto,
D. Rand and
F. Ferreira
Title:
Fine structure of hyperbolic diffeomorphisms
Additional book information:
Springer Monographs in Mathematics,
Springer-Verlag, Berlin, Heidelberg,
2009,
xvi+354 pp.,
ISBN 978-3-540-87524-6,
US$129.00,
hardcover
V. I. Arnol′d, Small denominators. I. Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 21–86 (Russian). MR 0140699
Rufus Bowen, Markov partitions for Axiom $\textrm {A}$ diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. MR 277003, DOI 10.2307/2373370
Elise E. Cawley, The Teichmüller space of an Anosov diffeomorphism of $T^2$, Invent. Math. 112 (1993), no. 2, 351–376. MR 1213107, DOI 10.1007/BF01232439
P. Coullet and C. Tresser, Itérations d’endomorphismes et groupe de renormalisation, J. Phys. Colloque C 539, C5–25 (1978).
Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 501179, DOI 10.1007/BF01020332
Edson de Faria, Welington de Melo, and Alberto Pinto, Global hyperbolicity of renormalization for $C^r$ unimodal mappings, Ann. of Math. (2) 164 (2006), no. 3, 731–824. MR 2259245, DOI 10.4007/annals.2006.164.731
V. Z. Grines and E. V. Zhuzhoma, Expanding attractors, Regul. Chaotic Dyn. 11 (2006), no. 2, 225–246. MR 2245079, DOI 10.1070/RD2006v011n02ABEH000347
Michael-Robert Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5–233 (French). MR 538680
Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR 0271991
R. de la Llave and R. Moriyón, Invariants for smooth conjugacy of hyperbolic dynamical systems. IV, Comm. Math. Phys. 116 (1988), no. 2, 185–192. MR 939045
Mikhail Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Ann. of Math. (2) 149 (1999), no. 2, 319–420. MR 1689333, DOI 10.2307/120968
Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347, DOI 10.1515/9781400865178
W. de Melo and A. A. Pinto, Rigidity of $C^2$ infinitely renormalizable unimodal maps, Comm. Math. Phys. 208 (1999), no. 1, 91–105. MR 1729879, DOI 10.1007/s002200050749
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 236383
R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR 1144770, DOI 10.1515/9781400882458
R. V. Plykin, Sources and sinks of $\textrm {A}$-diffeomorphisms of surfaces, Mat. Sb. (N.S.) 94(136) (1974), 243–264, 336 (Russian). MR 0356137
Clark Robinson, Dynamical systems, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. Stability, symbolic dynamics, and chaos. MR 1396532
A.G. Ruas, Atratores hiperbolicos de codimensão um e classes de isotopia de difeomorfismos PHd thesis, IMPA, 1982.
Ja. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen 2 (1968), no. 1, 64–89 (Russian). MR 0233038
Ya. G. Sinaĭ and K. M. Khanin, Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Uspekhi Mat. Nauk 44 (1989), no. 1(265), 57–82, 247 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 1, 69–99. MR 997684, DOI 10.1070/RM1989v044n01ABEH002008
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 169–203. MR 348794
Michael Shub and Dennis Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems 5 (1985), no. 2, 285–289. MR 796755, DOI 10.1017/S014338570000290X
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
Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 333–359 (French, with English summary). MR 777374
References
- V.I. Arnold, Small denominators I: on the maps of the circle into itself. Investijia Akad. Nauk. Math. 25, 1 (1961), 21–96. MR 0140699 (25:4113)
- R. Bowen, Markov partitions for Axiom $\textrm {A}$ diffeomorphisms. Amer. J. Math. 92 (1970), 725–747. MR 0277003 (43:2740)
- E. Cawley, The Teichmüller space of an Anosov diffeomorphism of $T^ 2$. Invent. Math. 112 (1993), no. 2, 351–376. MR 1213107 (94c:58141)
- P. Coullet and C. Tresser, Itérations d’endomorphismes et groupe de renormalisation, J. Phys. Colloque C 539, C5–25 (1978).
- M.J. Feigenbaum, Qualitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), 25–52. MR 0501179 (58:18601)
- E. de Faria, W. de Melo, A.A. Pinto. Global hyperbolicity of renormalization for $C^r$ unimodal mappings. Annals of Math., 164 (2006), 731–824. MR 2259245 (2009b:37071)
- V.Z. Grines, E.V. Zhuzhoma, Expanding attractors. Regul. Chaotic Dyn. 11 (2006), no. 2, 225–246. MR 2245079 (2007c:37015)
- M. Herman, Sur la conjugaison differentiable des difféomorphismes du cercle a des rotations, Publ. Math. IHES 49 (1979), 5–234. MR 538680 (81h:58039)
- M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, In: Global Analysis, Proc. Symp. in Pure Math., vol. XIV. American Math. Soc. (1970), 133–164. MR 0271991 (42:6872)
- R. de la Llave, R. Moriyón, Invariants for smooth conjugacy of hyperbolic dynamical systems. IV. Comm. Math. Phys. 116 (1988), no. 2, 185–192. MR 939045 (90h:58064)
- M. Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Annals of Math. 149 (1999), 319–420. MR 1689333 (2000d:37051)
- C. McMullen, Complex Dynamics and Renormalization, Annals of Math. Studies 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365 (96b:58097)
- C. McMullen, Renormalization and $3$-manifolds which fiber over the circle, Annals of Math. Studies 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347 (97f:57022)
- W. de Melo and A.A. Pinto, Smooth conjugacies between $C^2$ infinitely renormalizable quadratic maps with the same bounded type, Commun. Math. Phys. 208 (1999), 91–105. MR 1729879 (2001d:37043)
- W. de Melo & S. van Strien, One-dimensional dynamics, Springer-Verlag (1993). MR 1239171 (95a:58035)
- G.D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms. Inst. Hautes Études Sci. Publ. Math. 34 (1968) 53–104. MR 0236383 (38:4679)
- R.C. Penner, J.L. Harer, Combinatorics of train tracks. Annals of Mathematics Studies, 125. Princeton University Press, Princeton, NJ, 1992. xii+216 pp. MR 1144770 (94b:57018)
- R.V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces, Math. USSR Sb., 23 (1974), 223–253. MR 0356137 (50:8608)
- C. Robinson, Dynamical systems. Stability, symbolic dynamics, and chaos. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, (1995). MR 1396532 (97e:58064)
- A.G. Ruas, Atratores hiperbolicos de codimensão um e classes de isotopia de difeomorfismos PHd thesis, IMPA, 1982.
- Y. Sinai, Markov partitions and $U$-diffeomorphisms. Funkcional. Anal. i Prilozen 2 1968 no. 1, 64–89. [Russian] MR 0233038 (38:1361)
- Ya. G. Sinai, K. Khanin, Smoothness of conjugacies of diffeomorphisms of the circle with rotations. Uspekhi Mat. Nauk 44 (1989), no. 1(265), 57–82, 247 [Russian]; translation in Russian Math. Surveys 44 (1989), no. 1, 69–99 MR 997684 (90i:58183)
- S. Smale, Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967) 747–817. MR 0228014 (37:3598)
- R.F. Williams, Expanding attractors. Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 169–203. MR 0348794 (50:1289)
- M. Shub, D. Sullivan, Expanding endomorphisms of the circle revisited. Ergodic Theory Dynam. Systems 5 (1985), no. 2, 285–289. MR 796755 (87g:58104)
- D. Sullivan, Quasiconformal homeomorphisms and dynamics, topology and geometry. Proc. ICM-86, Berkeley, v. II 1216–1228. MR 934326 (90a:58160)
- D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, AMS Centennial Publications (1992). MR 1184622 (93k:58194)
- W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596 (89k:57023)
- J-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 333–359. MR 777374 (86j:58086)
Review Information:
Reviewer:
W. De Melo
Affiliation:
Instituto de Matemática Pura e. Aplicada
Email:
demelo@impa.br
Journal:
Bull. Amer. Math. Soc.
48 (2011), 131-136
DOI:
https://doi.org/10.1090/S0273-0979-2010-01284-2
Published electronically:
May 24, 2010
Review copyright:
© Copyright 2010
American Mathematical Society
