SELECTED PAPERS
Translated from Russian; 1966--1977
- Approximation of ergodic dynamical systems by periodic transformations (with A.M.Stepin), Soviet Math. Dokl 7, (1966), 1638-1641
- Approximations in ergodic theory (with A.M.Stepin), Russian Math. Surveys
22 (1967), 81-106
- Multiplicative Properties of Automorphisms, An unpublished chapter of the Ph.D. Thesis Some applications of the method of periodic approximations of dynamical systems to ergodic theory, Moscow State university, 1968.
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New examples in smooth ergodic theory. Ergodic diffeomorphisms (with D.V. Anosov), Trans. Moscow Math. Soc.
23 (1970), 1-35
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Ergodic flows generated by a system of weakly interacting oscillators (translation from Russian), Proceedings of the V International conference on non-linear oscillations
2 (1970), 216-221
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Minimal diffeomorphisms on principal S1-bundles (translation from Russian), Abstracts of the IV Topological Conference, Tbilisi , Mecniereba,
(1972), 63
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Invariant measures of flows on oriented surfaces, Soviet Math. Dokl. 14, (1973), 1104-1108
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Ergodic perturbations of degenerate integrable Hamiltonian systems Izv. Akad. Nauk SSSR Ser. Mat. 37, No. 3 (1973), 535-571
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Topological transitivity of billiards in polygons (with A.M. Zemlyakov) Math. Notes of the Academy of Sciences of the USSR, 18, (1975), 760-764;
errata 20 (1976), 1051
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Theory of dynamical systems and general transformation groups with invariant measure (with Ja. G. Sinai and A.M. Stepin) Mathematical Analysis,
13 Moscow, 1975, 129-261
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Monotone equivalence in ergodic theory, Math. USSR Izvestija 10 (1977), 104-157
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A conjecture about entropy AMS Translations 133 (1986), 91-108, published in Russian in 1977
1978-2016
SECTIONS AND NETS
- The special representation theorem for multi-dimensional group actions, Asterisque, 49, (1977), 117-140.
- Nets and Quasi-isometries (with D. Bernstein and G,D. Mostow), 1982.
HYPERBOLICITY, ENTROPY, GEODESIC FLOWS
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Bernoulli diffeomorphisms on surfaces Annals of mathematics 110 (1979), 529-547
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Bernoulli diffeomorphisms and group extensions of dynamical systems with non-zero characteristic exponents (with M. Brin and J. Feldman), Annals of mathematics, 113 (1981), 159-179
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Smooth non-Bernoulli K-automorphisms Invenetiones Math. 61 (1980), 291-300
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Lyapunov exponents, entropy and periodic points for diffeomorphisms Publ. Math. IHES 51 (1980), 137-173
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Entropy and closed geodesics Erg. Theory and Dynam. Systems 2 (Alexeyev memorial issue) (1982), 339-367
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Smooth models of Thurston's Pseudo-Anosov maps (with M. Gerber), Ann. Sci. Ecole Norm. Super. Paris 15, (1982), 173--204
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On local entropy (with M. Brin) Springer Lecture Notes in Math 1007 (1983), 30-38
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Non-uniform hyperbolicity and periodic orbits for smooth dynamical systems,Proc. Intern. Congress of Math. Warsaw 1983 2PWN-North Holland (1984), 1245-1254
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Random perturbations of transformations of an interval (with Yu. Kifer), J. Anal. Math., 47 (1986), 194-237
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Four applications of conformal equivalence to geometry and dynamics,Erg. Theory and Dynam. Systems, 8* (Conley memorial issue), (1988), 139-152.
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Differentiability and analyticity of topological entropy
for Anosov and geodesic flows (with G. Knieper, G. Pollicott and H. Weiss)
Invenetiones Math. 98 (1989), 581-597
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Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows
(with G. Knieper, and H. Weiss) Commun. Math. Phys. 138 (1991), 19-31
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Infinitesimal Lyapunov functions, invariant cone
families and stochastic properties of smooth dynamical systems (in collaboration with K. Burns),
Erg. Theory and Dynam. Systems, 14
(1994), 757-785
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Entropy and growth of expanding periodic orbits for
one-dimensional maps (with A. Mezhirov), Fund. Math.,
157 (1998),
245-254
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Fifty years of Entropy in Dynamics: 1958--2007 Journal of Modern Dynamics, 1, N4, (2007), 545--596.
INTERVAL EXCHANGE TRANSFORMATIONS, BILLIARDS, TWIST MAPS
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Interval exchange transformations and some special flows are not mixing, Israel J. Math.
35 (1980), 301-310
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Some remarks on Birkhoff and Mather twist map theorems,Erg. Theory and Dynam. Systems 2, 1982), No.2, 185-194
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The growth rate for the number of singular and periodic orbits for a polygonal billiard
Commun. Math. Phys. 111 (1987), 151-160
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Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians (with D. Bernstein)
Inventiones Math. 88(1987), 225-241
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Weakly mixing billiards (with E. Gutkin) Springer Lecture Notes in Math
1345 (1988) 163--176.
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Minimal orbits for small perturbations of completely integrable Hamiltonian systems, in Twist mappings and their applications Proc. Workshop, Minneapolis/MN (USA), 1989-90, IMA Vol. Math. Appl. 44 (1992) 183--199
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Caustics in inner and outer billiards
(with E. Gutkin),
Comm. Math. Phys., 173
(1995), 101-133
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Billiard table as a playground for a mathematician. (in Russian); in Studencheskie
chtenja v NMU, 2 Moscow MCCME (2001), 8--36.
(in English) ; Surveys in modern mathematics, 216--242, London Math. Soc. Lecture Note Ser., 321 , Cambridge Univ. Press, Cambridge, 2005. Improved English translation .
SPECTRAL THEORY, APPROXIMATION, COMBINATORIAL CONSTRUCTIONS
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Slow entropy type invariants and smooth realization of commuting
measure-preserving transformations (with J.-P. Thouvenot), Ann. Inst. H. Poincare,
33, N3 (1997),
323-338
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Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric
and smooth measure (with R. Gunesch), Disc. and Cont. Dyn. Sys.,
6 (2000),
61-83
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Cocycles, cohomology and combinatorial constructions
in ergodic theory (in collaboration with E. A. Robinson, Jr.), in Smooth Ergodic Theory and its
applications, Proc. Symp. Pure Math.,
69 (2001),
107-173;
also appears as the second part of Combinatorial constructions in ergodic theory and dynamics.
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Mixed spectrum
reparameterizations of linear flows on T2
(with B. Fayad and A. Windsor), Moscow
Mathematical Journal, 1, No. 4 (2001), 521-537
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Constructions
in elliptic dynamics (with B.Fayad), Erg. Theory and
Dynam. Systems,(Herman memorial issue), 24 (2004), 1477-1520
with an error in the published version corrected
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Spectral properties and combinatorial constructions in ergodic theory
(with J.-P.Thouvenot), Handbook in Dynamical systems, vol. 1B, Elsevier, (2005), 649-743
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Some new cases of realization of spectral multiplicity function
for ergodic transformations (with M. Lemanczyk), Fundamenta Mathematicae, 206 (2009), 185-215.
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Analytic uniquely ergodic volume preserving maps on odd spheres(with B. Fayad), Commentarii mathematici Helvetici, 89 N4, (2014), 963--977.
CHARACTERISTIC CLASSES AND DYNAMICS
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Ergodic theory and Weil measures for foliations (with S. Hurder),Annals of mathematics, 126, 1987, 221-275
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Differentiability, rigidity and Godbillon-Vey classes for Anosov flows (with S. Hurder),Publ. Math. I.H.E.S. 72, (1990), 5-61 ; unabridged preprint
COHOMOLOGY AND GEOMETRIC RIGIDITY
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Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature (with R. Feres),
Erg. Theory and Dynam. Systems 10 (1990), 657-670
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Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows (with R. Feres)
Erg. Theory and Dynam. Systems 9 (1989), 427-432
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Rigidity of symplectic Anosov diffeomorphisms on low dimensional tori (with L. Flaminio)
Erg. Theory and Dynam. Systems 11 (1991), 427-440
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Local rigidity for certain groups of toral automorphisms
(with J. Lewis),
Israel J. Math, 75
(1991), 203-241
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First cohomology of Anosov actions of higher rank abelian groups
and applications to rigidity (with R. Spatzier), Publ. Math. I.H.E.S., 79 (1994),
131-156, Errata
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Subelliptic estimates of polynomial differential operators and
applications to rigidity of abelian actions (with R. Spatzier), Math. Res. Letters,
1 (1994),
193-202
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Higher cohomology for Abelian groups of toral endomorphisms
(with S. Katok),
Erg. Theory and Dynam. Systems, 15
(1995), 569-592, Errata
-
The cohomology of expansive
Zd -actions by automorphisms of compact abelian
groups
(with K. Schmidt),
Pacific. J. Math., 170
(1995), 105-142
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Global rigidity results
for lattice actions on tori and new examples of
volume-preserving action
(with J. Lewis),
Israel J. Math, 93
(1996), 253-280
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Cocycle super-rigidity and rigidity for
lattice actions on tori
(with J. Lewis and R. Zimmer),
Topology, 35
(1996), 27-38
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Cocycle stability
for partially hyperbolic systems
(with A. Kononenko),
Math. Res. Letters, 3
(1996), 191-210
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Hyperbolic measures and
commuting maps in low dimension,
Disc. and Cont. Dyn. Sys, 2
(1996), 397-411
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Non-stationary normal forms and rigidity of
group actions (with R. Spatzier), ERA-AMS, 2, N3 (1996),
124-133
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Differential rigidity of Anosov actions of higher
rank Abelian groups and
algebraic lattice actions (with R. Spatzier), in Dynamical systems and related topics,
(Volume dedicated to D.V. Anosov), Proc. Steklov Math. Inst , 216 (1997),
287-314
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Normal forms and invariant geometric structures for dynamical systems with
invariant contracting foliations (with M. Guysinsky), Math. Res. Letters,
5 (1998),
149-163
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Non-Abelian cohomology of abelian
Anosov actions (with V. Nitica and A.Torok), Erg.
Theory and Dynam. Systems,
20 (2000),
259-288
- Higher cohomology for Abelian groups
of toral automorphisms II: the partially hyperbolic case, and corrigendum
(with S. Katok), Erg. Theory and
Dynam. Systems, 25 (2005), 1909-1917
- Periodic cycle functionals
and cocycle rigidity for certain partially hyperbolic Rk actions
(with D. Damjanovic), Discrete and Continuous Dynamical Systems , 13 (2005), 985-1005
- Rigidity of higher rank abelian cocycles with values in diffeomorphism groups
(with V. Nitica), Geometriae Dedicata 124, No1, (2007),
109-131.
- Local rigidity of restrictions of Weyl chamber flows
(with D. Damjanovic), Comptes rendus Mathematique, 344, N8 (2007), 503--508.
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Local rigidity of partially
hyperbolic actions. I. KAM method and Zk actions on the Torus (with D. Damjanovic), Annals of Mathematics, 172 (2010), 1805--1858
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Local rigidity of partially
hyperbolic actions.
II. The geometric method and restrictions of Weyl chamber flows on SL(n,,R)/\Gamma, (with D. Damjanovic), Int Math, Res. Notes, 2010; doi: 10.1093/imrn/rnq252.
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Rigidity of real-analytic actions of SL(n,Z) on
Tn : A case of realization of Zimmer program (with F. Rodriguez Hertz), Discrete and Continuous Dynamical Systems , 27 (2010), 609--615
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Local rigidity of homogeneous parabolic actions: I. A model case (with D. Damjanovic), Journal of Modern Dynamics, 5, N2, (2011)., 203--235.
MEASURE RIGIDITY
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Invariant measures for higher rank hyperbolic
Abelian actions (with R. Spatzier), Erg. Theory and
Dynam. Systems, 16 (1996),
751-778, Corrections to "Invariant measures for higher rank
hyperbolic Abelian actions", Erg. Theory and
Dynam. Systems, 18 (1998),
503-507
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Invariant measures for
actions of higher rank abelian groups (with B. Kalinin), in Smooth Ergodic Theory and its
applications, Proc. Symp. Pure Math.,
69 (2001),
593--637
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Measurable rigidity and disjointness
for
Zk actions by toral automorphisms (with B. Kalinin), Erg.
Theory and Dynam. Systems,
22 (2002),
507-523
-
Rigidity
of measurable structure for
Zd actions by automorphisms of a
torus (with S. Katok and K. Schmidt), Comm. Math. Helv., 77 (2002), 718-745
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Invariant measures on G/Γ for split
simple Lie
Groups G (with M. Einsiedler), Comm. Pure. Appl. Math., (Moser memorial
issue), 56 (2003), 1184-1221
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Rigidity of measures -- the high entropy case, and non-commuting foliations
(with M. Einsiedler), Israel Journal of Mathematics , 148 (2005), 169-238
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Invariant
measures and the set of exceptions to Littlewood's conjecture (with
M. Einsiedler and E. Lindenstrauss), Annals of Mathematics, 164 (2006), 513--560
- Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori
(with B. Kalinin), Journal of Modern Dynamics, 1, N1, (2007), 123--146 ; Errata
- Uniqueness of large invariant measures for Zk actions with Cartan homotopy data
(with F. Rodriguez Hertz), Journal of Modern Dynamics, 1, N2, (2007), 287--300 ; Errata
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New progress in nonuniform measure and cocycle rigidity (with B. Kalinin and F. Rodriguez Hertz), Electronic Research Announcements in Mathematical Sciences 15, (2008), 79--92.
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Nonuniform measure rigidity (with B. Kalinin and F. Rodriguez Hertz),
Annals of Mathematics, 174, (2011), 361--400
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Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher rank abelian groups (with F. Rodriguez Hertz), Journal of Modern Dynamics, 4, N3, (2010), 487--515.
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The Fried average entropy and slow entropy for actions of higher rank abelian groups
(with S. Katok and F. Rodriguez Hertz), Geometric and Functional Analysis, 24, N4, (2014), 1204--1228.
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Arithmeticity and topology of higher rank actions of Abelian groups (with F. Rodriguez Hertz) Journal of Modern Dynamics 10 (2016), 135--172.
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Non-uniform measure rigidity for Zk actions of symplectic type (with F. Rodriguez Hertz) in Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov, Contemporary mathematics, AMS, 692, 2017, 195--208.
UNPUBLISHED MATHEMATICAL WRITINGS
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Measure theory through dynamical eyes (with Vaughn Climenhaga)
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More about Birkhoff periodic orbits and Mather sets for twist maps
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Continuation of the preprint "More about Birkhoff periodic orbits and Mather sets for twist
maps"
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Katok
Last updated on July 6, 2017