Book Review
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Book Information:
Author:
G. A. Margulis
Title:
On some aspects of the theory of Anosov systems
Additional book information:
Springer Monographs in Mathematics, Springer,
Berlin,
2004,
138 pp.,
ISBN 3-540-40121-0,
$69.95$
D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
D. V. Anosov and Ja. G. Sinaĭ, Certain smooth ergodic systems, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 107–172 (Russian). MR 0224771
Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460. MR 339281, DOI 10.2307/2373793
Rufus Bowen, The equidistribution of closed geodesics, Amer. J. Math. 94 (1972), 413–423. MR 315742, DOI 10.2307/2374628
Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357–390. MR 1626749, DOI 10.2307/121012
David Fried, Meromorphic zeta functions for analytic flows, Comm. Math. Phys. 174 (1995), no. 1, 161–190. MR 1372805
G. A. Margulis, Certain measures that are connected with U-flows on compact manifolds, Funkcional. Anal. i Priložen. 4 (1970), no. 1, 62–76 (Russian). MR 0272984
William Parry and Mark Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math. (2) 118 (1983), no. 3, 573–591. MR 727704, DOI 10.2307/2006982
Mark Pollicott and Richard Sharp, Exponential error terms for growth functions on negatively curved surfaces, Amer. J. Math. 120 (1998), no. 5, 1019–1042. MR 1646052
M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math. 15 (1973), 92–114. MR 339282, DOI 10.1007/BF02771776
David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
David Ruelle, Dynamical zeta functions for piecewise monotone maps of the interval, CRM Monograph Series, vol. 4, American Mathematical Society, Providence, RI, 1994. MR 1274046, DOI 10.1090/crmm/004
Hans H. Rugh, The correlation spectrum for hyperbolic analytic maps, Nonlinearity 5 (1992), no. 6, 1237–1263. MR 1192517
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
Ja. G. Sinaĭ, Asymptotic behavior of closed geodesics on compact manifolds with negative curvature, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 1275–1296 (Russian). MR 0229178
Ja. G. Sinaĭ, Construction of Markov partitionings, Funkcional. Anal. i Priložen. 2 (1968), no. 3, 70–80 (Loose errata) (Russian). MR 0250352
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
- [A]
- D. V. Anosov. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. Math. 90 (1967). Transl. A. M. S. MR 0224110
- [A.Si]
- D. V. Anosov and Ya. G. Sinai. Certain smooth ergodic systems. Russ. Math. Surv. 22 No. 5 (1967), 103-167. MR 0224771
- [B.1]
- R. Bowen. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429-460. MR 0339281
- [B.2]
- R. Bowen. The equidistribution of closed geodesics. Amer. J. Math. 94 (1972), 413-423. MR 0315742
- [D]
- D. Dolgopyat. On decay of correlations in Anosov flows. Ann. of Math. 147 (1998), 357-390. MR 1626749
- [F]
- D. Fried. Meromorphic zeta functions for analytic flows. Comm. Math. Phys. 174 (1995), 161-190. MR 1372805
- [M]
- G. Margulis. Certain measures associated with U-flows on compact manifolds. Func. Anal. Appl. 4 (1969), 55-67. MR 0272984
- [P.P]
- W. Parry and M. Pollicott. An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. of Math. 118 (1983), 573-591. MR 0727704
- [P.S]
- M. Pollicott and R. Sharp. Exponential error terms for growth functions on negatively curved surfaces. Amer. J. Math. 120 (1998), 1019-1042. MR 1646052
- [Ra]
- M. Ratner. Markov partitions for Anosov flows on
-dimensional manifolds. Israel J. Math. 15 (1973), 92-114. MR 0339282
- [R.1]
- D. Ruelle. Thermodynamic Formalism. Addison Wesley, Reading, MA, 1978. MR 0511655
- [R.2]
- D. Ruelle. Dynamical zeta functions for piecewise monotone maps of the interval. CRM monograph series No. 4. A. M. S., 1994. MR 1274046
- [Ru]
- H. H. Rugh. The correlation spectrum for hyperbolic analytic maps. Nonlinearity 5 (1992), 1237-1263. MR 1192517
- [Se]
- A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20 (1956), 47-87. MR 0088511
- [Si.1]
- Ya. G. Sinai. The asymptotic behaviour of the number of closed geodesics on a compact manifold of negative curvature. Transl. A. M. S. 73 (1968), 227-250. MR 0229178
- [Si.2]
- Ya. G. Sinai. The construction of Markov partitions. Func. Anal. Appl. 2 (1968), 70-80. MR 0250352
- [Sm]
- S. Smale. Differentiable dynamical systems. Bull. A. M. S. 73 (1967), 747-817. MR 0228014
Review Information:
Reviewer:
William Parry
Affiliation:
University of Warwick
Email:
b.parry@warwick.ac.uk
Journal:
Bull. Amer. Math. Soc.
42 (2005), 257-261
Published electronically:
February 10, 2005
Review copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
