A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas
[Un contre-exemple à la décomposition lisse de Hodge pour des feuilletages généraux et à une classe de formules de trace dynamique]
Deninger, Christopher1 ; Singhof, Wilhelm2
1 WWU Münster, Mathematisches Institut, Einsteinstr. 62, 48149 Münster (Allemagne)
2 Universität Düsseldorf, Mathematisches Institut, Universitätsstr. 1, 40225 Düsseldorf (Allemagne)
Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 209-219.

Nous construisons un feuilletage à feuilles denses et de codimension deux sur la variété de Heisenberg tel que la décomposition de Hodge feuilletée ne soit pas vérifiée. Nous démontrons aussi qu'un certain type de formules de trace dynamique reliant les orbites périodiques aux traces sur la cohomologie feuilletée, ne sont pas vraies pour des flots arbitraires.

We construct a two dimensional foliation with dense leaves on the Heisenberg nilmanifold for which smooth leafwise Hodge decomposition does not hold. It is also shown that a certain type of dynamical trace formulas relating periodic orbits with traces on leafwise cohomologies does not hold for arbitrary flows.

DOI : 10.5802/aif.1821
Classification : 53C12, 37C27, 58J60
Keywords: Hodge decomposition, foliation, dynamical trace formula, nilmanifold
Mot clés : théorie de Hodge, formule de trace, feuilletage
Deninger, Christopher 1 ; Singhof, Wilhelm 2

1 WWU Münster, Mathematisches Institut, Einsteinstr. 62, 48149 Münster (Allemagne)
2 Universität Düsseldorf, Mathematisches Institut, Universitätsstr. 1, 40225 Düsseldorf (Allemagne) 
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     title = {A counterexample to smooth leafwise {Hodge} decomposition for general foliations and to a type of dynamical trace formulas},
     journal = {Annales de l'Institut Fourier},
     pages = {209--219},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Deninger, Christopher; Singhof, Wilhelm. A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas. Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 209-219. doi : 10.5802/aif.1821. http://www.numdam.org/articles/10.5802/aif.1821/

[AGH] L. Auslander; L. Green; F. Hahn Flows on homogeneous spaces, Ann. Math. Studies, 53, Princeton, 1963 | Zbl

[AK1] J.A. Álvarez López; Y. Kordyukov Long time behaviour of leafwise heat flow for Riemannian foliations (To appear in Compositio Math.) | Zbl

[AK2] J.A. Álvarez López; Y. Kordyukov Distributional Betti numbers of Lie Foliations (2000) (Manuscript)

[AT] J.A. Álvarez López; P. Tondeur Hodge decomposition along the leaves of a Riemannian foliation, J. Functional Analysis, Volume 99 (1991), pp. 443-458 | DOI | MR | Zbl

[CR] J.M. Cygan; L.F. Richardson Global solvability on compact nilmanifolds, Transactions AMS, Volume 301 (1987), pp. 343-373 | DOI | MR | Zbl

[D1] C. Deninger Some analogies between number theory and dynamical systems on foliated spaces, ICM (Doc. Math. J. DMV), Volume Extra Volume I (1998), pp. 23-46 | Zbl

[D2] C. Deninger On dynamical systems and their possible significance for Arithmetic Geometry II (2000) (Preprint) | MR | Zbl

[DS] C. Deninger; W. Singhof A note on dynamical trace formulas (2000) (Preprint) | MR | Zbl

[G] V. Guillemin Lectures on spectral theory of elliptic operators, Duke Math. J., Volume 44 (1977), pp. 485-517 | DOI | MR | Zbl

[Go] R. Goodman Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc., Volume 143 (1969), pp. 55-76 | DOI | MR | Zbl

[GS] V. Guillemin; S. Sternberg Geometric asymptotics, Math. Surveys, 14, Amer. Math. Soc., Providence, R.I, 1977 | MR | Zbl

[H] R. Howe On Frobenius reciprocity for unipotent algebraic groups over ℚ , Amer. J. of Math., Volume 93 (1971), pp. 163-172 | DOI | MR | Zbl

[MS] C.C. Moore; C. Schochet Global Analysis on foliated spaces, MSRI Publications, 9, Springer, 1988 | MR | Zbl

[P] S.J. Patterson On Ruelle's zeta function, Israel Math. Conf. Proc., Volume 3 (1990), pp. 163-184 | MR | Zbl

[R1] L.F. Richardson Decomposition of the L 2 -space of a general compact nilmanifold, Amer. J. of Math., Volume 93 (1971), pp. 173-190 | DOI | MR | Zbl

[R2] L.F. Richardson Global solvability on compact Heisenberg manifolds, Transactions AMS (1982), pp. 309-317 | DOI | MR | Zbl

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