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 3–56 (Mi umn5788)  

This article is cited in 118 scientific papers (total in 118 papers)

Lectures on the entropy theory of measure-preserving transformations

V. A. Rokhlin

Full text: PDF file (5805 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 1967, 22:5, 1–52

Bibliographic databases:

UDC: 519.9
MSC: 37A05, 28D05, 37A35, 28D20, 37A25
Received: 14.11.1966

Citation: V. A. Rokhlin, “Lectures on the entropy theory of measure-preserving transformations”, Uspekhi Mat. Nauk, 22:5(137) (1967), 3–56; Russian Math. Surveys, 22:5 (1967), 1–52

Citation in format AMSBIB
\by V.~A.~Rokhlin
\paper Lectures on the entropy theory of measure-preserving transformations
\jour Uspekhi Mat. Nauk
\yr 1967
\vol 22
\issue 5(137)
\pages 3--56
\jour Russian Math. Surveys
\yr 1967
\vol 22
\issue 5
\pages 1--52

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. A. G. Kushnirenko, “On metric invariants of entropy type”, Russian Math. Surveys, 22:5 (1967), 53–61  mathnet  crossref  mathscinet  zmath
    2. A. B. Katok, A. M. Stepin, “Metric properties of measure preserving homeomorphisms”, Russian Math. Surveys, 25:2 (1970), 191–220  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. A. M. Vershik, “Continuum of pairwise nonisomorphic diadic sequences”, Funct. Anal. Appl., 5:3 (1971), 182–184  mathnet  crossref  mathscinet  zmath
    6. K. L. Volkovysskii, Ya. G. Sinai, “Ergodic properties of an ideal gas with an infinite number of degrees of freedom”, Funct. Anal. Appl., 5:3 (1971), 185–187  mathnet  crossref  mathscinet  zmath
    7. A. M. Stepin, “On entropy invariants of decreasing sequences of measurable partitions”, Funct. Anal. Appl., 5:3 (1971), 237–240  mathnet  crossref  mathscinet  zmath
    8. V. G. Sharapov, “Markov endomorphisms of Lebesgue space”, Funct. Anal. Appl., 5:3 (1971), 246–249  mathnet  crossref  mathscinet  zmath
    9. T. Schwartzbauer, “Approximation of measure-preserving transformations”, Math Systems Theory, 6:3 (1972), 312  crossref  mathscinet
    10. W. K. Hayman, S. M. Rudolfer, “On the relation between some metric invariants for Markov shifts”, Periodica Mathematica Hungarica Journal of the J anos Bolyai Mathematical Society, 2:1-4 (1972), 133  crossref  mathscinet  zmath
    11. Ya. G. Sinai, “Gibbs measures in ergodic theory”, Russian Math. Surveys, 27:4 (1972), 21–69  mathnet  crossref  mathscinet  zmath
    12. Alan Saleski, “On induced transformations of Bernoulli shifts”, Math Systems Theory, 7:1 (1973), 83  crossref  mathscinet  zmath
    13. Ernst Eberlein, “Einbettung von Strömungen in FunktionenrÄume durch Erzeuger vom endlichen Typ”, Z Wahrscheinlichkeitstheorie verw Gebiete, 27:4 (1973), 277  crossref  mathscinet  zmath
    14. A. B. Katok, “Ergodic perturbations of degenerate integrable Hamiltonian systems”, Math. USSR-Izv., 7:3 (1973), 535–571  mathnet  crossref  mathscinet  zmath
    15. B. A. Rubshtein, “O posledovatelnostyakh izmerimykh razbienii, porozhdennykh endomorfizmami prostranstva Lebega”, UMN, 28:1(169) (1973), 247–248  mathnet  mathscinet  zmath
    16. Michael D Weiss, “Topological entropy of Chebyshev polynomials and related real mappings”, Journal of Mathematical Analysis and Applications, 43:3 (1973), 816  crossref
    17. Ernst Eberlein, “A generator theorem for flows”, Bol Soc Bras Mat, 5:1 (1974), 45  crossref  mathscinet  zmath
    18. FranÇois Ledrappier, “Principe variationnel et systèmes dynamiques symboliques”, Z Wahrscheinlichkeitstheorie verw Gebiete, 30:3 (1974), 185  crossref  mathscinet  zmath
    19. Isaac Meilijson, “Mixing properties of a class of skew-products”, Isr J Math, 19:3 (1974), 266  crossref  mathscinet
    20. Michael D. Weiss, “Algebraic and other entropies of group endomorphisms”, Math Systems Theory, 8:3 (1974), 243  crossref  mathscinet
    21. M. I. Brin, Ya. B. Pesin, “Partially hyperbolic dynamical systems”, Math. USSR-Izv., 8:1 (1974), 177–218  mathnet  crossref  mathscinet  zmath
    22. L. A. Bunimovich, “On billiards close to dispersing”, Math. USSR-Sb., 23:1 (1974), 45–67  mathnet  crossref  mathscinet  zmath
    23. Yu. I. Kifer, “On small random perturbations of some smooth dynamical systems”, Math. USSR-Izv., 8:5 (1974), 1083–1107  mathnet  crossref  mathscinet
    24. D. S. Ornshtein, “Primeneniya ergodicheskoi teorii k teorii veroyatnostei”, UMN, 30:3(183) (1975), 125–146  mathnet  mathscinet  zmath
    25. T. V. Volkova, “O maksimalnykh kommutativnykh regulyarnykh podalgebrakh v faktore tipa II$_1$”, UMN, 30:6(186) (1975), 163–164  mathnet  mathscinet  zmath
    26. Alan Saleski, “A note on a differentiation type property of the entropy of an automorphism”, Advances in Mathematics, 15:2 (1975), 157  crossref
    27. M. Casartelli, L. Galgani, A. Scotti, “Numerical computations on a stochastic parameter related to the Kolmogorov entropy”, Phys Rev A, 13:5 (1976), 1921  crossref  adsnasa
    28. Ya. B. Pesin, “Characteristic Lyapunov exponents and smooth ergodic theory”, Russian Math. Surveys, 32:4 (1977), 55–114  mathnet  crossref  mathscinet  zmath
    29. A. B. Katok, “Monotone equivalence in ergodic theory”, Math. USSR-Izv., 11:1 (1977), 99–146  mathnet  crossref  mathscinet  zmath  isi
    30. Ya. B. Pesin, “Geodesic flows on closed Riemannian manifolds without focal points”, Math. USSR-Izv., 11:6 (1977), 1195–1228  mathnet  crossref  mathscinet  zmath
    31. B. M. Gurevich, “Perfect partitions for ergodic flows”, Funct. Anal. Appl., 11:3 (1977), 179–182  mathnet  crossref  mathscinet  zmath
    32. A. M. Vershik, “The action of $PLS(2, \mathbb Z)$ in $\mathbb R^1$ is approximable”, Russian Math. Surveys, 33:1 (1978), 221–222  mathnet  crossref  mathscinet  zmath
    33. Ya. G. Sinai, “Ergodic properties of the Lorentz gas”, Funct. Anal. Appl., 13:3 (1979), 192–202  mathnet  crossref  mathscinet  zmath
    34. J F C van Velsen, J Phys A Math Gen, 14:7 (1981), 1621  crossref  mathscinet  zmath  adsnasa  isi
    35. John C. Kieffer, “A direct proof that vwb processes are closed in the
      $$\bar d$$
      -metric”, Isr J Math, 41:1-2 (1982), 154  crossref  mathscinet  zmath  isi
    36. 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
    37. M. Ju. Ljubich, “Entropy properties of rational endomorphisms of the Riemann sphere”, Ergod Th Dynam Sys, 3:3 (1983)  crossref  mathscinet
    38. F. Ledrappier, “Propriétés Ergodiques Des Mesures De Sinaï”, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 59:1 (1984), 163  crossref  mathscinet  zmath
    39. D. B. Sten’kin, “Preservation of Statistical Properties for Subsequences of Random Sequences”, Theory Probab Appl, 28:4 (1984), 752  mathnet  crossref  mathscinet  zmath  isi
    40. Roger Butler, Klaus Schmidt, “An information cocycle for groups of non-singular transformations”, Z Wahrscheinlichkeitstheorie verw Gebiete, 69:3 (1985), 347  crossref  mathscinet  zmath  isi
    41. Frank Novak, Ronnie Kosloff, David J. Tannor, Andras Lorincz, Duane D. Smith, Stuart A. Rice, “Wave packet evolution in isolated pyrazine molecules: Coherence triumphs over chaos”, J Chem Phys, 82:3 (1985), 1073  crossref  isi
    42. Uhland Burkart, “Orbit entropy in noninvertible mappings”, J Austral Math Soc, 40:1 (1986), 95  crossref  mathscinet  zmath  isi
    43. Lloyd Demetrius, “Random spin models and chemical kinetics”, J Chem Phys, 87:12 (1987), 6939  crossref  mathscinet  isi
    44. 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
    45. A. N. Livshits, “On the spectra of adic transformations of Markov compacta”, Russian Math. Surveys, 42:3 (1987), 222–223  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    46. P. Ferrero, B. Schmitt, “Produits aléatoires d'opérateurs matrices de transfert”, Probab Theory Relat Fields, 79:2 (1988), 227  crossref  mathscinet  zmath  isi
    47. Meishan Zhao, Stuart A. Rice, “Comment concerning the optimum control of transformations in an unbounded quantum system”, J Chem Phys, 95:4 (1991), 2465  crossref  isi
    48. L. -S. Young, “Decay of correlations for certain quadratic maps”, Comm Math Phys, 146:1 (1992), 123  crossref  mathscinet  zmath  adsnasa  isi
    49. Liu Peidong, Qian Min, “Ruelle's inequality for the entropy of diffusion processes”, Acta Math Sinica, 9:2 (1993), 212  crossref  mathscinet  zmath
    50. Huyi Hu, “Some ergodic properties of commuting diffeomorphisms”, Ergod Th Dynam Sys, 13:1 (1993)  crossref  mathscinet  zmath
    51. Huyi Hu, Lai-Sang Young, “Nonexistence of SBR measures for some diffeomorphisms that are ‘Almost Anosov’”, Ergod Th Dynam Sys, 15:1 (1995)  crossref  mathscinet  zmath
    52. Michael Boshernitzan, Isaac Kornfeld, “Interval translation mappings”, Ergod Th Dynam Sys, 15:5 (1995), 821  crossref
    53. R. Kenyon, Y. Peres, “Measures of full dimension on affine-invariant sets”, Ergod Th Dynam Sys, 16:2 (1996)  crossref  mathscinet  zmath
    54. B. M. Gurevich, S. V. Savchenko, “Thermodynamic formalism for countable symbolic Markov chains”, Russian Math. Surveys, 53:2 (1998), 245–344  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    55. O. N. Ageev, “Dynamical systems with arbitrary spectrum multiplicity function”, Russian Math. Surveys, 53:5 (1998), 1079–1081  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    56. S. V. Savchenko, “Special Flows Constructed From Countable Topological Markov Chains”, Funct. Anal. Appl., 32:1 (1998), 32–41  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    57. O. N. Ageev, “The spectral multiplicity function and geometric representations of interval exchange transformations”, Sb. Math., 190:1 (1999), 1–28  mathnet  crossref  crossref  mathscinet  zmath  isi
    58. O. N. Ageev, “On the spectral multiplicity function for dynamical systems”, Math. Notes, 65:4 (1999), 516–519  mathnet  crossref  crossref  mathscinet  zmath  isi
    59. Hiroki Sumi, Nonlinearity, 13:4 (2000), 995  crossref  mathscinet  zmath  isi
    60. Mirko Degli Esposti, Gianluigi Del Magno, Marco Lenci, Nonlinearity, 13:4 (2000), 1275  crossref  mathscinet  zmath  isi
    61. M. I. Gordin, “Double extensions of dynamical systems and constructing mixing filtrations”, Journal of Mathematical Sciences (New York), 99:2 (2000), 1053  crossref  mathscinet
    62. Christian Bonatti, Marcelo Viana, “SRB measures for partially hyperbolic systems whose central direction is mostly contracting”, Isr J Math, 115:1 (2000), 157  crossref  mathscinet  zmath  isi
    63. A. M. Vershik, “Dynamic theory of growth in groups: Entropy, boundaries, examples”, Russian Math. Surveys, 55:4 (2000), 667–733  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    64. Lindenstrauss, E, “Bernoulli convolutions and an intermediate value theorem for entropies of K-partitions”, Journal D Analyse Mathematique, 87 (2002), 337  crossref  mathscinet  zmath  isi
    65. Kaimanovich, VA, “Boundary and entropy of space homogeneous Markov chains”, Annals of Probability, 30:1 (2002), 323  crossref  mathscinet  zmath  isi  elib
    66. Pei-Dong Liu, “Ruelle Inequality Relating Entropy, Folding Entropy and Negative Lyapunov Exponents”, Commun. Math. Phys, 240:3 (2003), 531  crossref
    67. Erschler, A, “Liouville property for groups and manifolds”, Inventiones Mathematicae, 155:1 (2004), 55  crossref  mathscinet  zmath  isi  elib
    68. A. M. Vershik, “Towards the definition of metric hyperbolicity”, Mosc. Math. J., 5:3 (2005), 721–737  mathnet  mathscinet  zmath
    69. A. G. Kachurovskii, “The entropy brick of an automorphism of a Lebesgue space”, Math. Notes, 80:6 (2006), 885–887  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    70. A. M. Vershik, A. D. Gorbul'skii, “Scaled entropy of filtrations of $\sigma$-fields”, Theory Probab. Appl., 52:3 (2008), 493–508  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    71. S. V. Tikhonov, “A complete metric in the set of mixing transformations”, Sb. Math., 198:4 (2007), 575–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    72. Dooley A.H., Mortiss G., “The Critical Dimensions of Hamachi Shifts”, Tohoku Math. J., 59:1 (2007), 57–66  crossref  isi
    73. Mona Khare, Shraddha Roy, “Conditional Entropy and the Rokhlin Metric on an Orthomodular Lattice with Bayessian State”, Int J Theor Phys, 47:5 (2008), 1386  crossref  mathscinet  zmath  isi
    74. ELI GLASNER, XIANGDONG YE, “Local entropy theory”, Ergod Th Dynam Sys, 2008, 1  crossref  mathscinet  isi
    75. Mona Khare, Shraddha Roy, “Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State”, Commun. Theor. Phys, 50:3 (2008), 551  crossref
    76. Dooley A.H., Golodets V.Ya., Rudolph D.J., Sinel'shchikov S.D., “Non-Bernoulli systems with completely positive entropy”, Ergodic Theory Dynam Systems, 28:1 (2008), 87–124  isi
    77. Hu H., “Equilibriums of Some Non-Holder Potentials”, Trans. Am. Math. Soc., 360:4 (2008), 2153–2190  isi
    78. Alexandre I. Danilenko, Daniel J. Rudolph, “Conditional entropy theory in infinite measure and a question of Krengel”, Isr J Math, 172:1 (2009), 93  crossref  mathscinet  zmath  isi
    79. Yves Benoist, Jean-François Quint, “Mesures stationnaires et fermés invariants des espaces homogènes”, Comptes Rendus Mathematique, 347:1-2 (2009), 9  crossref
    80. Chunyan Mu, David Clark, “An Interval-based Abstraction for Quantifying Information Flow”, Electronic Notes in Theoretical Computer Science, 253:3 (2009), 119  crossref
    81. Eugen Mihailescu, “Physical Measures for Multivalued Inverse Iterates Near Hyperbolic Repellors”, J Statist Phys, 2010  crossref  isi
    82. VÍTOR ARAÚJO, ALEXANDER I. BUFETOV, “A large deviations bound for the Teichmüller flow on the moduli space of abelian differentials”, Ergod Th Dynam Sys, 2010, 1  crossref
    83. FRANÇOIS LEDRAPPIER, JIAN-SHENG XIE, “Vanishing transverse entropy in smooth ergodic theory”, Ergod Th Dynam Sys, 2010, 1  crossref
    84. EUGEN MIHAILESCU, “On a class of stable conditional measures”, Ergod Th Dynam Sys, 2010, 1  crossref
    85. Eugen Mihailescu, “Local Geometry and Dynamical Behavior on Folded Basic Sets”, J Stat Phys, 2010  crossref
    87. Janvresse E., Meyerovitch T., Roy E., de la Rue T., “Poisson Suspensions and Entropy for Infinite Transformations”, Trans. Am. Math. Soc., 362:6 (2010), 3069–3094  isi
    88. A. M. Vershik, “Scailing entropy and automorphisms with pure pointspectrum”, St. Petersburg Math. J., 23:1 (2012), 75–91  mathnet  crossref  mathscinet  zmath  isi  elib
    89. Yves Benoist, Jean-François Quint, “Mesures stationnaires et fermés invariants des espaces homogènes”, Ann. Math, 174:2 (2011), 1111  crossref
    90. A. N. Gorbachev, A. M. Stepin, “On the structure of invariant measures for set-valued maps”, Sb. Math., 202:9 (2011), 1285–1302  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    91. A. I. Bufetov, B. M. Gurevich, “Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials”, Sb. Math., 202:7 (2011), 935–970  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    92. Eugen Mihailescu, “On some coding and mixing properties for a class of chaotic systems”, Monatsh Math, 2011  crossref
    93. Miklós Rédei, Charlotte Werndl, “On the history of the isomorphism problem of dynamical systems with special regard to von Neumann’s contribution”, Arch. Hist. Exact Sci, 2011  crossref
    95. Piotr Oprocha, Guohua Zhang, “Dimensional entropy over sets and fibres”, Nonlinearity, 24:8 (2011), 2325  crossref
    96. Eugen Mihailescu, “Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms”, DCDS-a, 32:7 (2012), 2485  crossref
    97. L. I. Rodina, “Invariantnye i statisticheski slabo invariantnye mnozhestva upravlyaemykh sistem”, Izv. IMI UdGU, 2012, no. 2(40), 3–164  mathnet
    98. JIAN-SHENG XIE, “A new entropy formula of Ledrappier–Young type for linear toral dynamics”, Ergod. Th. Dynam. Sys, 2012, 1  crossref
    99. Eugen Mihailescu, Mariusz Urbański, “Entropy Production for a Class of Inverse SRB Measures”, J Stat Phys, 2012  crossref
    100. Dooley A.H., Golodets V.Ya., “On the Entropy of Actions of Nilpotent Lie Groups and their Lattice Subgroups”, Ergod. Theory Dyn. Syst., 32:Part 2 (2012), 535–573  crossref  isi
    101. Huang W., Li H., Ye X., “Family Independence for Topological and Measurable Dynamics”, Trans. Am. Math. Soc., 364:10 (2012), 5209–5242  isi
    102. Jean-Pierre Conze Guy Cohen, “The CLT for rotated ergodic sums and related processes”, DCDS-A, 33:9 (2013), 3981  crossref
    103. Eugene Kagan, Irad Ben-Gal, “A Group-Testing Algorithm with Online Informational Learning”, IIE Transactions, 2013, 1305290631  crossref
    104. Eugen Mihailescu, Mariusz Urbański, “Measure-theoretic degrees and topological pressure for non-expanding transformations”, Journal of Functional Analysis, 2014  crossref
    105. Huang W., Ye X., Zhang G., “Lowering Topological Entropy Over Subsets Revisited”, Trans. Am. Math. Soc., 366:8 (2014), 4423–4442  isi
    106. MICHIHIRO HIRAYAMA, NAOYA SUMI, “On the ergodicity of hyperbolic Sinaǐ–Ruelle–Bowen measures: the constant unstable dimension case”, Ergod. Th. Dynam. Sys, 2015, 1  crossref
    107. J. Math. Sci. (N. Y.), 209:6 (2015), 826–829  mathnet  crossref
    108. Marko Budišić, Jean-Luc Thiffeault, “Finite-time braiding exponents”, Chaos, 25:8 (2015), 087407  crossref
    109. P. B. Zatitskiy, “Scaling entropy sequence: invariance and examples”, J. Math. Sci. (N. Y.), 209:6 (2015), 890–909  mathnet  crossref
    110. D. Z. Arov, “To the history of the appearance of the notion of $\epsilon$-entropy of an authomorphism of a Lebesque space and $(\varepsilon,T)$-entropy of a dynamical system with continuous time”, J. Math. Sci. (N. Y.), 215:6 (2016), 677–692  mathnet  crossref  mathscinet
    111. B. M. Gurevich, “Toward the history of dynamical entropy: comparing two definitions”, J. Math. Sci. (N. Y.), 215:6 (2016), 693–699  mathnet  crossref  mathscinet
    112. Gurevich B.M., “a Lower Estimate of the Entropy of An Automorphism and Maximum Entropy Conditions For and Invariant Measure of a Suspension Flow Over a Markov Shift”, 91, no. 2, 2015, 186–188  crossref  isi
    113. A. V. Alpeev, “Announce of an entropy formula for a class of Gibbs measures”, J. Math. Sci. (N. Y.), 224:2 (2017), 171–175  mathnet  crossref  mathscinet
    114. A. M. Vershik, “The theory of filtrations of subalgebras, standardness, and independence”, Russian Math. Surveys, 72:2 (2017), 257–333  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    115. B. M. Gurevich, “Entropy of a stationary process and entropy of a shift transformation in its sample space”, Problems Inform. Transmission, 53:2 (2017), 103–113  mathnet  crossref  isi  elib
    116. B. M. Gurevich, “Affinity of the Arov Entropy”, Funct. Anal. Appl., 52:3 (2018), 178–185  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    117. Austin T., “Measure Concentration and the Weak Pinsker Property”, Publ. Math. IHES, 128:1 (2018), 1–119  crossref  mathscinet  zmath  isi  scopus
    118. A. M. Vershik, “Asimptotika razbieniya kuba na simpleksy Veilya i kodirovanie skhemy Bernulli”, Funkts. analiz i ego pril., 53:2 (2019), 11–31  mathnet  crossref  elib
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:2031
    Full text:857
    First page:1

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