General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Uspekhi Mat. Nauk:

Personal entry:
Save password
Forgotten password?

Uspekhi Mat. Nauk, 1967, Volume 22, Issue 5(137), Pages 107–172 (Mi umn5792)  

This article is cited in 108 scientific papers (total in 110 papers)

Some smooth ergodic systems

D. V. Anosov, Ya. G. Sinai

Full text: PDF file (7397 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 1967, 22:5, 103–167

Bibliographic databases:

Document Type: Article
UDC: 519.9
MSC: 37C10, 37C40, 37C05, 53D25, 37C75
Received: 14.11.1966

Citation: D. V. Anosov, Ya. G. Sinai, “Some smooth ergodic systems”, Uspekhi Mat. Nauk, 22:5(137) (1967), 107–172; Russian Math. Surveys, 22:5 (1967), 103–167

Citation in format AMSBIB
\by D.~V.~Anosov, Ya.~G.~Sinai
\paper Some smooth ergodic systems
\jour Uspekhi Mat. Nauk
\yr 1967
\vol 22
\issue 5(137)
\pages 107--172
\jour Russian Math. Surveys
\yr 1967
\vol 22
\issue 5
\pages 103--167

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. Smeil, “Differentsiruemye dinamicheskie sistemy”, UMN, 25:1(151) (1970), 113–185  mathnet  mathscinet
    2. Ya. G. Sinai, “Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards”, Russian Math. Surveys, 25:2 (1970), 137–189  mathnet  crossref  mathscinet  zmath
    3. G. A. Margulis, “Certain measures associated with $U$-flows on compact manifolds”, Funct. Anal. Appl., 4:1 (1970), 55–67  mathnet  crossref  mathscinet  zmath
    4. E. I. Dinaburg, “On the relations among various entropy characteristics of dynamical systems”, Math. USSR-Izv., 5:2 (1971), 337–378  mathnet  crossref  mathscinet  zmath
    5. N. M. Pukhov, D. S. Chernavskii, “On phase instability in the case of scattering by a random potential”, Theoret. and Math. Phys., 7:2 (1971), 487–494  mathnet  crossref
    6. Philip Hartman, “The stable manifold of a point of a hyperbolic map of a Banach space”, Journal of Differential Equations, 9:2 (1971), 360  crossref
    7. Ya. G. Sinai, “Gibbs measures in ergodic theory”, Russian Math. Surveys, 27:4 (1972), 21–69  mathnet  crossref  mathscinet  zmath
    8. A. B. Kramli, “Geodezicheskie potoki na kompaktnykh rimanovykh poverkhnostyakh bez fokalnykh tochek”, UMN, 27:5(167) (1972), 245–246  mathnet  mathscinet  zmath
    9. Spotswood D. Stoddard, Joseph Ford, “Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System”, Phys Rev A, 8:3 (1973), 1504  crossref  adsnasa
    10. M. Ratner, “Markov partitions for anosov flows onn-dimensional manifolds”, Isr J Math, 15:1 (1973), 92  crossref  mathscinet  zmath
    11. L. A. Bunimovich, Ya. G. Sinai, “On a fundamental theorem in the theory of dispersing billiards”, Math. USSR-Sb., 19:3 (1973), 407–423  mathnet  crossref  mathscinet  zmath
    12. Ya. B. Pesin, “On the existence of invariant fiberings for a diffeomorphism of a smooth manifold”, Math. USSR-Sb., 20:2 (1973), 213–222  mathnet  crossref  mathscinet  zmath
    13. A. B. Katok, “Ergodic perturbations of degenerate integrable Hamiltonian systems”, Math. USSR-Izv., 7:3 (1973), 535–571  mathnet  crossref  mathscinet  zmath
    14. M. I. Brin, Ya. B. Pesin, “Chastichno giperbolicheskie dinamicheskie sistemy”, UMN, 28:3(171) (1973), 169–170  mathnet  mathscinet  zmath
    15. A. B. Kramli, “Potoki biedrov na trekhmernom mnogoobrazii otritsatelnoi krivizny”, UMN, 28:4(172) (1973), 219–220  mathnet  mathscinet  zmath
    16. M. Ratner, “Anosov flows with gibbs measures are also Bernoullian”, Isr J Math, 17:4 (1974), 380  crossref  mathscinet  zmath
    17. M. I. Brin, Ya. B. Pesin, “Partially hyperbolic dynamical systems”, Math. USSR-Izv., 8:1 (1974), 177–218  mathnet  crossref  mathscinet  zmath
    18. L. A. Bunimovich, “On a class of special flows”, Math. USSR-Izv., 8:1 (1974), 219–232  mathnet  crossref  mathscinet  zmath
    19. V. T. Perekrest, “Ob eksponentsialnom peremeshivanii v U-sistemakh”, UMN, 29:1(175) (1974), 181–182  mathnet  mathscinet  zmath
    20. L. A. Bunimovich, “On billiards close to dispersing”, Math. USSR-Sb., 23:1 (1974), 45–67  mathnet  crossref  mathscinet  zmath
    21. D. S. Ornshtein, “Primeneniya ergodicheskoi teorii k teorii veroyatnostei”, UMN, 30:3(183) (1975), 125–146  mathnet  mathscinet  zmath
    22. C. F. Driscoll, T. M. O'Neil, “Explanation of Instabilities Observed on a Fermi-Pasta-Ulam Lattice”, Phys Rev Letters, 37:2 (1976), 69  crossref  adsnasa
    23. Ya. B. Pesin, “Families of invariant manifolds corresponding to nonzero characteristic exponents”, Math. USSR-Izv., 10:6 (1976), 1261–1305  mathnet  crossref  mathscinet  zmath
    24. Ya. B. Pesin, “Characteristic Lyapunov exponents and smooth ergodic theory”, Russian Math. Surveys, 32:4 (1977), 55–114  mathnet  crossref  mathscinet  zmath
    25. Ya. B. Pesin, “Geodesic flows on closed Riemannian manifolds without focal points”, Math. USSR-Izv., 11:6 (1977), 1195–1228  mathnet  crossref  mathscinet  zmath
    26. Marina Ratner, “Horocycle flows are loosely Bernoulli”, Isr J Math, 31:2 (1978), 122  crossref  mathscinet  zmath
    27. M. Ratner, “Bernoulliflows over maps of the interval”, Isr J Math, 31:3-4 (1978), 298  crossref  mathscinet  zmath
    28. J.F.C. Van Velsen, “On the Riemann curvature of conservative systems in classical mechanics”, Physics Letters A, 67:5-6 (1978), 325  crossref
    29. Marina Ratner, “The cartesian square of the horocycle flow is not loosely Bernoulli”, Isr J Math, 34:1-2 (1979), 72  crossref  mathscinet  zmath  isi
    30. Ya. G. Sinai, “Ergodic properties of the Lorentz gas”, Funct. Anal. Appl., 13:3 (1979), 192–202  mathnet  crossref  mathscinet  zmath
    31. Marc Mangel, “Uniform treatment of fluctuations at critical points”, Physica A: Statistical Mechanics and its Applications, 97:3 (1979), 597  crossref
    32. J F C van Velsen, J Phys A Math Gen, 13:3 (1980), 833  crossref  mathscinet  zmath  adsnasa  isi
    33. L. A. Bunimovich, Ya. G. Sinai, “Markov Partitions for dispersed billiards”, Comm Math Phys, 78:2 (1980), 247  crossref  mathscinet  zmath  adsnasa  isi  elib
    34. Kenneth G. Kay, “Numerical study of intramolecular vibrational energy transfer: Quantal, classical, and statistical behavior”, J Chem Phys, 72:11 (1980), 5955  crossref  mathscinet  adsnasa  isi  elib
    35. Ya. B. Pesin, “Geodesic flows with hyperbolic behaviour of the trajectories and objects connected with them”, Russian Math. Surveys, 36:4 (1981), 1–59  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    36. R. Englman, “Relaxation dynamics of excited impurities in solids”, Chemical Physics, 58:2 (1981), 227  crossref
    37. Robert J. Zimmer, “Ergodic theory, semisimple Lie groups, and foliations by manifolds of negative curvature”, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 55:1 (1982), 37  crossref  mathscinet  zmath
    38. M. V. Jacobson, “Invariant measures for some one-dimensional attractors”, Ergod Th Dynam Sys, 2:3-4 (1982)  crossref  mathscinet  zmath
    39. Ya. B. Pesin, Ya. G. Sinai, “Gibbs measures for partially hyperbolic attractors”, Ergod Th Dynam Sys, 2:3-4 (1982)  crossref  mathscinet  zmath
    40. Fran¸ois Ledrappier, Jean-Marie Strelcyn, “A proof of the estimation from below in Pesin's entropy formula”, Ergod Th Dynam Sys, 2:2 (1982)  crossref
    41. N. I. Chernov, “Structure of transversal leaves in multidimensional semidispersing billiards”, Funct. Anal. Appl., 16:4 (1982), 270–280  mathnet  crossref  mathscinet  zmath  isi
    42. Ronald F. Fox, “Long-time tails and diffusion”, Phys Rev A, 27:6 (1983), 3216  crossref  mathscinet  isi
    43. Kenneth G. Kay, “Toward a comprehensive semiclassical ergodic theory”, J Chem Phys, 79:6 (1983), 3026  crossref  mathscinet  adsnasa  isi
    44. William Parry, “Bowen's equidistribution theory and the Dirichlet density theorem”, Ergod Th Dynam Sys, 4:1 (1984)  crossref  mathscinet
    45. Gérard G. Emch, Sungpyo Hong, “Time-like geodesic flows on Lorentz manifolds”, Ergod Th Dynam Sys, 7:2 (1987)  crossref  mathscinet
    46. Ya. G. Sinai, N. I. Chernov, “Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls”, Russian Math. Surveys, 42:3 (1987), 181–207  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    47. F. Ledrappier, “Ergodic properties of Brownian motion on covers of compact negatively-curve manifolds”, Bol Soc Bras Mat, 19:1 (1988), 115  crossref  mathscinet  zmath
    48. Maciej P. Wojtkowski, “A system of one dimensional balls with gravity”, Comm Math Phys, 126:3 (1990), 507  crossref  mathscinet  zmath  isi
    49. S E Troubetzkoy, Nonlinearity, 3:3 (1990), 947  crossref  mathscinet  zmath  adsnasa  isi
    50. L. A. Bunimovich, Ya. G. Sinai, N. I. Chernov, “Statistical properties of two-dimensional hyperbolic billiards”, Russian Math. Surveys, 46:4 (1991), 47–106  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    51. V. V. Solodov, “Topological topics in dynamical systems theory”, Russian Math. Surveys, 46:4 (1991), 107–134  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    52. F.T. Farrell, L.E. Jones, “Foliated control without radius of injectivity restrictions”, Topology, 30:2 (1991), 117  crossref
    53. Ya. B. Pesin, “Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties”, Ergod Th Dynam Sys, 12:1 (1992)  crossref  mathscinet  zmath
    54. V. I. Bakhtin, “A direct method of constructing an invariant measure on a hyperbolic attractor”, Russian Acad. Sci. Izv. Math., 41:2 (1993), 207–227  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    55. E. A. Sataev, “Invariant measures for hyperbolic maps with singularities”, Russian Math. Surveys, 47:1 (1992), 191–251  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    56. R Markarian, Nonlinearity, 6:5 (1993), 819  crossref  mathscinet  zmath  adsnasa  isi
    57. V. P. Maslov, “On an integral equation of the form $u(x)=F(x)+\int G(x,\xi)u_+^{(n-2)/2}(\xi) d\xi/\int u_+^{(n-2)/2}(\xi) d\xi$ for $n=2$ and $n=3$”, Math. Notes, 55:3 (1994), 302–311  mathnet  crossref  mathscinet  zmath  isi
    58. V. P. Maslov, “On the Integral Equation $u(x)=F(x)+\int G(x,\xi)u_+^{k/2}(\xi) d\xi/\int u_+^{k/2}(\xi) d\xi$”, Funct. Anal. Appl., 28:1 (1994), 33–41  mathnet  crossref  mathscinet  zmath  isi
    59. V. P. Maslov, “Semiclassical asymptotics of the eigenfunctions of the Schrödinger-Hartree equation. New form of classical self-consistent field”, Theoret. and Math. Phys., 99:1 (1994), 484–493  mathnet  crossref  mathscinet  zmath  isi
    60. Gregory Gaspari, “The Arnold cat map on prime lattices”, Physica D: Nonlinear Phenomena, 73:4 (1994), 352  crossref
    61. Roberto Markarian, “Statistical properties of dynamical systems with singularities”, J Statist Phys, 80:5-6 (1995), 1207  crossref  mathscinet  zmath  isi
    62. V. V. Kozlov, “Integral invariants of the Hamilton equations”, Math. Notes, 58:3 (1995), 938–947  mathnet  crossref  mathscinet  zmath  isi
    63. Ch. Dellago, H. A. Posch, “Lyapunov instability in a system of hard disks in equilibrium and nonequilibrium steady states”, Phys Rev E, 53:2 (1996), 1485  crossref  adsnasa  isi
    64. András Krámli, “Decay of correlations in one and two dimensions”, J Statist Phys, 83:1-2 (1996), 167  crossref  mathscinet  zmath  isi
    65. S. P. Novikov, L. A. Bunimovich, A. M. Vershik, B. M. Gurevich, E. I. Dinaburg, G. A. Margulis, V. I. Oseledets, S. A. Pirogov, K. M. Khanin, N. N. Chentsova, “Yakov Grigor'evich Sinai (on his sixtieth birthday)”, Russian Math. Surveys, 51:4 (1996), 765–778  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    66. V. V. Ryzhikov, “Intertwinings of tensor products, and the stochastic centralizer of dynamical systems”, Sb. Math., 188:2 (1997), 237–263  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    67. V. I. Arnol'd, A. A. Bolibrukh, R. V. Gamkrelidze, V. P. Maslov, E. F. Mishchenko, S. P. Novikov, Yu. S. Osipov, Ya. G. Sinai, A. M. Stepin, L. D. Faddeev, “Dmitrii Viktorovich Anosov (on his 60th birthday)”, Russian Math. Surveys, 52:2 (1997), 437–445  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    68. G Contreras, R Iturriaga, H Sánchez-Morgado, Nonlinearity, 11:2 (1998), 355  crossref  mathscinet  zmath  adsnasa  isi
    69. Marek Szydłowski, Zdzislaw Golda, “The generalized local instability criterion and approximation method”, Chaos, Solitons & Fractals, 9:1-2 (1998), 105  crossref  elib
    70. George Bozis, Grigoris Pavliotis, Inverse Probl, 15:1 (1999), 141  crossref  mathscinet  zmath  isi
    71. Sérgio Fenley, Lee Mosher, “Quasigeodesic flows in hyperbolic 3-manifolds”, Topology, 40:3 (2001), 503  crossref
    72. Marlies Gerber, Boris Hasselblatt, Daniel Keesing, “The Riccati Equation: Pinching of Forcing and Solutions”, Experimental Mathematics, 12:2 (2003), 129  crossref
    73. Dolgopyat, D, “Limit theorems for partially hyperbolic systems”, Transactions of the American Mathematical Society, 356:4 (2004), 1637  crossref  mathscinet  zmath  isi  elib
    74. Steven Lansel, Mason A. Porter, Leonid A. Bunimovich, “One-particle and few-particle billiards”, Chaos, 16:1 (2006), 013129  crossref  mathscinet  zmath  isi
    75. Michihiro Hirayama, Yakov Pesin, “Non-absolutely continuous foliations”, Isr J Math, 160:1 (2007), 173  crossref  mathscinet  zmath  isi
    76. Pesin, Y, “Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents”, Regular & Chaotic Dynamics, 12:5 (2007), 476  crossref  mathscinet  adsnasa  isi
    77. NURLAN S. DAIRBEKOV, GABRIEL P. PATERNAIN, “Rigidity properties of Anosov optical hypersurfaces”, Ergod Th Dynam Sys, 2008, 1  crossref  isi
    78. Balint P., Toth I.P., “Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards”, Ann Henri Poincaré, 9:7 (2008), 1309–1369  crossref  isi
    79. Ulysse Serres, “On Zermelo-Like Problems: Gauss–Bonnet Inequality and E. Hopf Theorem”, J Dyn Control Syst, 2009  crossref  mathscinet  zmath  isi
    80. E. A. Sataev, “Invariant measures for singular hyperbolic attractors”, Sb. Math., 201:3 (2010), 419–470  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    81. Pesin Ya., Climenhaga V., “Open Problems in the Theory of Non-Uniform Hyperbolicity”, Discrete and Continuous Dynamical Systems, 27:2 (2010), 589–607  crossref  zmath  isi
    82. Mikko Stenlund, “A Strong Pair Correlation Bound Implies the CLT for Sinai Billiards”, J Statist Phys, 2010  crossref
    83. Pilar Herreros, James Vargo, “Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field”, J Geom Anal, 2010  crossref
    84. Katrin Gelfert, Adilson E. Motter, “(Non)Invariance of Dynamical Quantities for Orbit Equivalent Flows”, Comm Math Phys, 2010  crossref
    85. E. A. Sataev, “Stokhasticheskie svoistva singulyarno giperbolicheskikh attraktorov”, Nelineinaya dinam., 6:1 (2010), 187–206  mathnet  elib
    86. Liverani C., “On the Work and Vision of Dmitry Dolgopyat”, J. Mod. Dyn., 4:2 (2010), 211–225  crossref  isi
    87. Chengbo Li, “A Note on Hyperbolic Flows in Sub-Riemannian Structures with Transverse Symmetries”, Acta Appl Math, 2011  crossref
    88. Nazaruk E.M., Romanovskaya A.M., “Ob eksponentsialnoi dikhotomii reshenii zadachi koshi dlya sistemy differentsialno-raznostnykh uravnenii s postoyannymi koeffitsientami”, Vestnik omskogo universiteta, 2011, no. 4, 45–49  elib
    89. Mário Bessa, “On $C^1$ -Generic Chaotic Systems in Three-Manifolds”, Qual. Theory Dyn. Syst, 2012  crossref
    90. Romanovskii R.K., Nazaruk E.M., “Spektralnyi kriterii eksponentsialnoi dikhotomii dlya lineinoi avtonomnoi sistemy funktsionalno-differentsialnykh uravnenii”, Doklady Akademii nauk vysshei shkoly Rossiiskoi Federatsii, 2012, no. 1, 19–27  elib
    91. Massimo Tessarotto, Claudio Cremaschini, Marco Tessarotto, “On the conditions of validity of the Boltzmann equation and Boltzmann H-theorem”, Eur. Phys. J. Plus, 128:3 (2013)  crossref
    92. Massimo Cencini, Francesco Ginelli, “Lyapunov analysis: from dynamical systems theory to applications”, J. Phys. A: Math. Theor, 46:25 (2013), 250301  crossref
    93. Mikko Stenlund, Lai-Sang Young, Hongkun Zhang, “Dispersing Billiards with Moving Scatterers”, Commun. Math. Phys, 2013  crossref
    94. W.N.. Yessen, “On the Spectrum of 1D Quantum Ising Quasicrystal”, Ann. Henri Poincaré, 2013  crossref
    95. R. K. Romanovskii, E. M. Nazaruk, “On the Dichotomy of Linear Autonomous Systems of Functional-Differential Equations”, Math. Notes, 95:1 (2014), 116–121  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    96. Massimo Tessarotto, Claudio Cremaschini, “First-principle proof of the modified collision boundary conditions for the hard-sphere system”, Physics Letters A, 2014  crossref
    97. Massimo Tessarotto, Claudio Cremaschini, “The Master kinetic equation for the statistical treatment of the Boltzmann-Sinai classical dynamical system”, Eur. Phys. J. Plus, 129:7 (2014)  crossref
    98. L.A. Bunimovich, “Short- and long-term forecast for chaotic and random systems (50 years after Lorenz's paper)”, Nonlinearity, 27:9 (2014), R51  crossref
    99. Miaohua Jiang, “Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one”, DCDS-A, 35:3 (2014), 967  crossref
    100. Massimo Tessarotto, Claudio Cremaschini, “Modified BBGKY hierarchy for the hard-sphere system”, Eur. Phys. J. Plus, 129:11 (2014)  crossref
    101. Araujo V., Galatolo S., Pacifico M.J., “Statistical Properties of Lorenz-Like Flows, Recent Developments and Perspectives”, Int. J. Bifurcation Chaos, 24:10 (2014), 1430028  crossref  isi
    102. Massimo Tessarotto, Claudio Cremaschini, “Theory of collisional invariants for the Master kinetic equation”, Physics Letters A, 2015  crossref
    103. Massimo Tessarotto, Claudio Cremaschini, “Axiomatic foundations of entropic theorems for hard-sphere systems”, Eur. Phys. J. Plus, 130:5 (2015)  crossref
    104. S.N.. Simić, “On the sub-Riemannian geometry of contact Anosov flows”, J. Topol. Anal, 2015, 1  crossref
    105. S.N.. SIMIĆ, “Global cross sections for Anosov flows”, Ergod. Th. Dynam. Sys, 2015, 1  crossref
    106. Romanovskii R.K., Nazaruk E.M., “Dichotomy of Solutions of Functional-Differential Equations in the Sobolev Space”, 51, no. 4, 2015, 464–476  crossref  isi
    107. G. K. Savvidi, “Anosov C-systems and random number generators”, Theoret. and Math. Phys., 188:2 (2016), 1155–1171  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    108. V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, “On the structure of the ambient manifold for Morse–Smale systems without heteroclinic intersections”, Proc. Steklov Inst. Math., 297 (2017), 179–187  mathnet  crossref  crossref  mathscinet  isi  elib
    109. T. V. Dudnikova, “O neravnovesnykh sostoyaniyakh kristallicheskoi reshetki”, Preprinty IPM im. M. V. Keldysha, 2018, 015, 26 pp.  mathnet  crossref
    110. S. P. Kuznetsov, “Khaos i giperkhaos geodezicheskikh potokov na mnogoobraziyakh s kriviznoi, otvechayuschikh mekhanicheski svyazannym rotatoram: primery i chislennoe issledovanie”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:4 (2018), 565–581  mathnet  crossref  elib
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:1179
    Full text:429
    First page:7

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019