$ and $\Hcal^1=\Ucal/d\Ucal = \ZZ_p \oplus \ZZ_p$. \\ Work on finding Halmos invariants in ergodic theory continued for example in \cite{AkCh65} and \cite{Ste71}. \\ The link of cohomology of dynamical systems with algebraic group cohomology was established in \cite{FeMo77}. It was motivated by cohomology constructions of Westman for virtual groups \cite{Wes69} who generalized a construction of C.C. Moore in 1964. The representation theory of Mackey's virtual groups \cite{Mackey} was also a motivation for \cite{Kir67}, because Mackey had proposed to call elements of $H^1(G,U(N))$ 'unitary representations of a virtual subgroup' of $G$. The term "virtual subgroup" has now been soaked up by "groupoids" and is a central in noncommutative mathematics" (see \cite{Wei96} for a review). \\ For more information on orbit cohomology, we refer to the second chapter in \cite{Schmidt}. \\ The first cohomolocal group appears at different other places in other function spaces. They are especially important for the structural stability of dynamical systems: when conjugating two dynamical systems $T$, $\tilde{T}=T+f$ with a conjugating map $\phi=Id+h$, the unknown function $h$ has to satisfy the functional equation $$ T(x)+h(T(x))=T(x+h(x))+f(x+h(x)) \; .$$ In the case $T:\TT^1 \rightarrow \TT^1, x \mapsto x+\alpha$, this equation can be written as $h(x+\alpha) -h(x) =f(x+h(x))$ which has the linearisation the homological equation $h(x+\alpha)-h(x)=f(x)$. Solving this equation in a space of analytic, periodic functions is important in Arnold's proof \cite{Arnold} of the theorem, that an analytic diffeomorphism of the circle $\tilde{T}$ with Diophantine rotation number can be conjugated to the rigid rotation $T$ if the two diffeomorphisms $T,\tilde{T}$ are close enough to each other (see \cite{Her79} for the global solution). \\ For Anosov $\ZZ$ actions, cohomology questions can be reduced to the cohomology on the periodic orbits. See \cite{Liv72} in the H\"older and $C^1$ case, \cite{Ll+86} in the $C^{\infty}$ case following work of Guillemin and Kazhdan and in the realanalytic case \cite{Lla97}. \\ The cohomology in the H\"older category is also relevant in statistical mechanics \cite{Bowen,Ruelle}. \\ Because in the measurable category, the cohomology groups are independent of the dynamical system, one usually takes an ergodic group translation on the torus. The topic has applications for representation theory of noncompact nonabelian Lie groups \cite{Mer85,BaMe86,Bag88}, especially when the coefficient group is the circle $U(1)$ or other unitary groups $U(n)$. When the coefficient group is $\ZZ_2$, one deals with the cohomology of measurable sets which first appeared in \cite{Kir67}. The group is relevant for Lyapunov exponent of ${\rm SL}(2,\RR)$-cocycles \cite{Kni91} because cohomology determines whether switching of stable and unstable manifolds of a cocycle makes the Lyapunov exponent vanish. \\ Research on the first cohomology group for $\ZZ^d$ Anosov actions started in \cite{KaSp94} and is an active research topic (see \cite{Schmidt95} section~30 for some review on cohomological rigidity). \\ Acknowledgements. \\ The theorem in this paper was obtained as a Taussky-Todd instructor at Caltech. It was first bundeled together with results on Schr\"odinger operators \cite{Kni95}. My thanks go to Jean Pierre Conze who told me about \cite{Der93} and \cite{Kir67} in 1993, to Jack Feldman, who let me know in 1994 about Depauw's work, to J\'er\^ome Depauw for sending me a copy of his theses in 1995 and to Greg Hjorth, who showed me in 1995, how useful and beautyful descriptive set theory can be. While preparing the present version at UT, I gratefully acknowledge the support of the Swiss National Science foundation. \begin{thebibliography}{10} \bibitem{KaSp94} R.J.~Spatzier A.~Katok. \newblock First cohomology of {A}nosov actions of higher rank abelian groups and applications to rigidity. \newblock {\em Inst. Hautes \'Etudes Sci. Publ. Math.}, 79:131--156, 1994. \bibitem{AkCh65} M.A. Akcoglu and R.V. Chacon. \newblock Generalized eigenvalues of automorphisms. \newblock {\em Proc. Amer. Math. Soc.}, 16:676--680, 1965. \bibitem{Ano73} D.V. Anosov. \newblock On the additive functional homology equation connected with an ergodic rotation of the circle. \newblock {\em Izv. Akad. Nauk. SSSR}, 37:1257--1271, 1973. \bibitem{Anz51} H.~Anzai. \newblock Ergodic skew product transformations on the torus. \newblock {\em Osaka Math. J.}, 3:83--99, 1951. \bibitem{Arnold} V.~Arnold. \newblock {\em Geometrical methods in the theory of ordinary differential equations}, volume 250 of {\em Springer Grundlehren}. \newblock Springer Verlag, New York, 1988. \bibitem{Asch95} M.~Aschbacher. \newblock Personal communication. \newblock 1995. \bibitem{Bag88} L.~Baggett. \newblock On circle valued cocycles of an ergodic measure preserving transformation. \newblock {\em Israel J. Math.}, 61:29--38, 1988. \bibitem{BaMe86} L.~Baggett and K.Merrill. \newblock Representation of the mautner group and cocycles of an irrational rotation. \newblock {\em Mich. Math. J.}, 33:221--229, 1986. \bibitem{Bowen} R.~Bowen. \newblock {\em Equilibrium States and the ergodic theory of Anosov diffeomorphisms}, volume 470 of {\em Lecture Notes in Mathematics}. \newblock Springer Verlag, Berlin, 1975. \bibitem{Con+81} A.~Connes, J.Feldman, and B.Weiss. \newblock An amenable equivalence relation is generated by a single transformation. \newblock {\em Ergod. Th. Dyn. Sys.}, 1:431--450, 1981. \bibitem{CFS} I.P. Cornfeld, S.V. Fomin, and Ya.G. Sinai. \newblock {\em Ergodic Theory}, volume 115 of {\em {Grundlehren} der mathematischen {Wissenschaften} in {Einzeldarstellungen}}. \newblock Springer Verlag, 1982. \bibitem{Lla97} R.~de~la Llave. \newblock Analytic regularity of solutions of {L}ivsic's cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems. \newblock {\em Ergod. Th. Dyn. Sys.}, 17:649--662, 1997. \bibitem{Ll+86} R.~de~la Llave, J.M. Marco, and R.~Moriy{\'o}n. \newblock Canonical perturbation theory of {A}nosov systems and regularity results for the {L}iv\v sic cohomology equation. \newblock {\em AOM}, 123:537--611, 1986. \bibitem{DeP94} J.~DePauw. \newblock Th\'eor\`emes ergodiques pour cocycle de degr\'e 2. \newblock Th\`ese de doctorat, Universit\'e de Bretagne Occidentale, 1994. \bibitem{Der93} J.-M. Derrien. \newblock Crit\`eres d'ergodicit\'e de cocycles en escalier. {Exemples}. \newblock {\em C.\ R.\ Acad.\ Sc.\ Paris}, 316:73--76, 1993. \bibitem{Eck46} B.~Eckmann. \newblock Der {C}ohomologie-{R}ing einer beliebigen {G}ruppe. \newblock {\em CMH}, 18:232--282, 1946. \bibitem{EiMc47} S.~Eilenberg and S.McLane. \newblock Cohomology theory in abstract groups. \newblock {\em Annals of Mathematics}, 48:51--78, 1947. \bibitem{FeMo77} J.~Feldman and C.Moore. \newblock Ergodic equivalence relations, cohomology and von {N}eumann algebras {I,II}. \newblock {\em Trans. Am. Math. Soc.}, 234:289--359, 1977. \bibitem{Halmos} P.~Halmos. \newblock {\em Lectures on ergodic theory}. \newblock The mathematical society of {Japan}, 1956. \bibitem{Her79} M.~Herman. \newblock Sur la conjugaison {diff\'erentiable des diff\'eomorphismes} du cercle {\`a} des rotations. \newblock {\em Pub. I.H.E.S.}, 49:5--233, 1979. \bibitem{Hil02} D.~Hilbert. \newblock Mathematische probleme. \newblock In {\em Gesammelte Abhandlungen {III}}. Berlin 1935, 1902. \bibitem{Hjo95} G.~Hjorth. \newblock Personal communication. \newblock 1995. \bibitem{KaKa95} A.~Katok and S.~Katok. \newblock Higher cohomology for abelian groups of toral automorphisms. \newblock {\em Ergod. Th. Dyn. Sys.}, 15:569--592, 1995. \bibitem{Kat67} A.B. Katok. \newblock Spectral properties of dynamical systems with an integral invariant on the torus. \newblock {\em Functional Anal. Appl.}, 1:296--305, 1967. \bibitem{Kechris} A.S. Kechris. \newblock {\em Classical Descriptive Set Theory}, volume 156 of {\em Graduate Texts in Mathematics}. \newblock Springer-Verlag, Berlin, 1994. \bibitem{Kir67} A.A. Kirillov. \newblock Dynamical systems, factors and representations of groups. \newblock {\em Russ. Math. Surveys}, 22:63--75, 1967. \bibitem{Kni91} O.~Knill. \newblock The upper {Lyapunov} exponents of {$SL(2,R)$} cocycles: discontinuity and the problem of positivity. \newblock In J.P.~Eckmann L.~Arnold, H.~Crauel, editor, {\em Lyapunov Exponents}, pages 86--97. Springer-Verlag, Berlin, Heidelberg, NewYork, 1991. \newblock {Lecture Notes in Mathematics}, 1486. \bibitem{Kni93} O.~Knill. \newblock Spectral, ergodic and cohomological problems in dynamical systems. \newblock {PhD Theses, ETH Z\"urich}, 1993. \bibitem{Kni95} O.~Knill. \newblock Discrete random electromagnetic {Laplacians}. \newblock (unpublished preprint available in the Mathematical Physics Preprint Archive $mp_arc$ document $95-195$), 1995. \bibitem{KniI} O.~Knill. \newblock Random {Schr\"odinger} operators arising from lattice gauge fields {I}: existence and density of states. \newblock {\em J. of Math. Physics}, 40:5495--5510, 1999. \bibitem{Kol53} A.N. Kolmogorov. \newblock On dynamical systems with an integral invariant on the torus. \newblock {\em Dokl. Akad. Nauk. SSSR, Ser. Mat.}, 93:763--766, 1953. \newblock Translated in: {\em Selected Works of A.N.~Kolmogorov, Volume~1: Mathematics and Mechanics}, V.M. Tikhomirov, editor, Kluwer Academic Publishers, 1991, pp. 344--348. \bibitem{MacLane} S.~Mac Lane. \newblock {\em Homology}. \newblock Classics in Mathematics. Springer-Verlag, Berlin, 1995. \newblock Reprint of the 1975 edition. \bibitem{Lin78} D.A. Lind. \newblock Products of coboundaries for commuting nonsingular automorphisms. \newblock {\em Z. Wahrsch. verw. Gebiete}, 43:135--139, 1978. \bibitem{Liv72} A.~Livsic. \newblock Cohomology of dynamical systems. \newblock {\em Izv. Akad. Nauk SSSR}, 36:1278--1301, 1972. \bibitem{Mackey} G.W. Mackey. \newblock {\em Induced representations of groups and quantum mechanics}. \newblock W. A. Benjamin, Inc., New York-Amsterdam, 1968. \bibitem{Mer85} K.D. Merrill. \newblock Cohomology of step functions under irrational rotations. \newblock {\em Israel J. Math.}, 52:320--340, 1985. \bibitem{Mos62} J.~Moser. \newblock On invariant curves of area-preserving mappings of an annulus. \newblock {\em {Nachr. Akad. Wiss. G\"ottingen Math.-Phys. Kl. II}}, 1962:1--20, 1962. \bibitem{PT} W.~Parry and S.Tuncel. \newblock {\em Classification problems in ergodic theory}. \newblock London Mathematical Society Lecture Note Series, 67. Cambridge University press, 1982. \bibitem{Ruelle} D.~Ruelle. \newblock {\em Thermodynamic Formalism. {The} Mathematical Structures of Classical Equilibrium Statistical Mechanics}, volume~5 of {\em Encyclopedia of mathematics and its applications}. \newblock Addison-Wesley Publishing Company, London, 1978. \bibitem{Schmidt} K.~Schmidt. \newblock {\em Algebraic ideas in ergodic theory}, volume~76 of {\em Regional conference Series in mathematics}. \newblock AMS, Providence, 1989. \bibitem{Schmidt95} K.~Schmidt. \newblock {\em Dynamical systems of Algebraic origin}, volume 128 of {\em Progress in Mathematics}. \newblock {Birkh\"auser Verlag}, Basel, 1995. \bibitem{Shk67} M.D. Shklover. \newblock On classical dynamical systems on the torus with continuous spectrum. \newblock {\em Izv. Vyssh. Uchebn. Zaved. Mat.}, 10:113--124, 1967. \newblock In Russian. \bibitem{Ste71} A.M. Stepin. \newblock Cohomologies of automorphism groups of a {Lebesgue} space. \newblock {\em Functional Anal. Appl.}, 5:167--168, 1971. \bibitem{Neu32} J.~von Neuman. \newblock Zur operatorenmethode in der mechanik. \newblock {\em Annals of Mathematics}, 33:587--642, 1932. \bibitem{Wei96} A.~Weinstein. \newblock Groupoids: unifying internal and external symmetry. {A} tour through some examples. \newblock {\em Notices of the AMS}, 43:744--752, 1996. \bibitem{Wes69} J.~J. Westman. \newblock Cohomology for ergodic groupoids. \newblock {\em Trans. Am. Math. Soc.}, 146:465--471, 1969. \bibitem{Win45} A.~Wintner. \newblock The linear difference equation of first order for angular variables. \newblock {\em Duke Math.\ J.}, 12:445--449, 1945. \end{thebibliography} \end{document}