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Izv. Akad. Nauk SSSR Ser. Mat., 1972, Volume 36, Issue 6, Pages 1296–1320 (Mi izv2373)  

This article is cited in 26 scientific papers (total in 26 papers)

Cohomology of dynamical systems

A. N. Livshits


Abstract: In this paper criteria for the cohomological nullity of functions on phase spaces of various dynamical systems (U-systems, topological Markov chains, Smale systems) with coefficients in certain groups are formulated and proved, and applications of these criteria are studied. (In the simplest case of a transformation $T\colon M\to M$, the cohomological nullity of a real function $f$ means that $f(x)=g(Tx)-g(x)$.)

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English version:
Mathematics of the USSR-Izvestiya, 1972, 6:6, 1278–1301

Bibliographic databases:

UDC: 517.9
MSC: Primary 58F99, 57F99; Secondary 34C20, 60J10
Received: 27.12.1971

Citation: A. N. Livshits, “Cohomology of dynamical systems”, Izv. Akad. Nauk SSSR Ser. Mat., 36:6 (1972), 1296–1320; Math. USSR-Izv., 6:6 (1972), 1278–1301

Citation in format AMSBIB
\Bibitem{Liv72}
\by A.~N.~Livshits
\paper Cohomology of dynamical systems
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1972
\vol 36
\issue 6
\pages 1296--1320
\mathnet{http://mi.mathnet.ru/izv2373}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=334287}
\zmath{https://zbmath.org/?q=an:0252.58007}
\transl
\jour Math. USSR-Izv.
\yr 1972
\vol 6
\issue 6
\pages 1278--1301
\crossref{https://doi.org/10.1070/IM1972v006n06ABEH001919}


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    Citing articles on Google Scholar: Russian citations, English citations
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    2. William A. Veech, “Periodic points and invariant pseudomeasures for toral endomorphisms”, Ergod Th Dynam Sys, 6:3 (1986)  crossref  mathscinet
    3. R. Llave, R. Moriyón, “Invariants for smooth conjugacy of hyperbolic dynamical systems. IV”, Comm Math Phys, 116:2 (1988), 185  crossref  mathscinet  zmath  adsnasa
    4. Mahesh G. Nerurkar, “On the construction of smooth ergodic skew-products”, Ergod Th Dynam Sys, 8:2 (1988)  crossref  mathscinet  zmath
    5. S. Hurder, A. Katok, “Differentiability, rigidity and Godbillon-Vey classes for Anosov flows”, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 72:1 (1990), 5  crossref  mathscinet  zmath
    6. Zaqueu Coelho, “On the asymptotic range of cocyles for shifts of finite type”, Ergod Th Dynam Sys, 13:2 (1993)  crossref  mathscinet  zmath
    7. Anatole Katok, Ralf J. Spatzier, “First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity”, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 79:1 (1994), 131  crossref  mathscinet  zmath
    8. William Parry, “Instances of cohomological triviality and rigidity”, Ergod Th Dynam Sys, 15:4 (1995)  crossref  mathscinet
    9. S. V. Savchenko, “Cohomological Inequalities for Finite Topological Markov Chains”, Funct. Anal. Appl., 33:3 (1999), 236–238  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. Matthew Nicol, Ian Melbourne, Peter Ashwin, Nonlinearity, 14:2 (2001), 275  crossref  mathscinet  zmath  isi
    11. Mark Pollicott, Howard Weiss, “Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a Potential?)”, Commun. Math. Phys, 240:3 (2003), 457  crossref
    12. J. AARONSON, M. DENKER, O. SARIG, R. ZWEIMÜLLER, “APERIODICITY OF COCYCLES AND CONDITIONAL LOCAL LIMIT THEOREMS”, Stoch. Dyn, 04:01 (2004), 31  crossref
    13. D. I. Dolgopyat, “Averaging and invariant measures”, Mosc. Math. J., 5:3 (2005), 537–576  mathnet  crossref  mathscinet  zmath
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    15. Martin Bridgeman, “Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space”, Geom Topol, 14:2 (2010), 799  crossref
    16. Boris Kalinin, “Livšic Theorem for matrix cocycles”, Ann. Math, 173:2 (2011), 1025  crossref
    17. Boris Kalinin, Victoria Sadovskaya, “Cocycles with one exponent over partially hyperbolic systems”, Geom Dedicata, 2012  crossref
    18. Victoria Sadovskaya, “Cohomology of $GL(2,\mathbb{R})$-valued cocycles over hyperbolic systems”, DCDS-A, 33:5 (2012), 2085  crossref
    19. MARIO PONCE, “Hyperbolization of cocycles by isometries of the Euclidean space”, Ergod. Th. Dynam. Sys, 2012, 1  crossref
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    21. A. SAMBARINO, “Hyperconvex representations and exponential growth”, Ergod. Th. Dynam. Sys, 2013, 1  crossref
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    24. VICTORIA SADOVSKAYA, “Cohomology of fiber bunched cocycles over hyperbolic systems”, Ergod. Th. Dynam. Sys, 2014, 1  crossref
    25. Maja Resman, “ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations”, Nonlinearity, 27:12 (2014), 3005  crossref
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