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.
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.
Keywords: Hodge decomposition, foliation, dynamical trace formula, nilmanifold
Mot clés : théorie de Hodge, formule de trace, feuilletage
Deninger, Christopher 1; Singhof, Wilhelm 2
@article{AIF_2001__51_1_209_0, author = {Deninger, Christopher and Singhof, Wilhelm}, 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}, volume = {51}, number = {1}, year = {2001}, doi = {10.5802/aif.1821}, zbl = {0997.58017}, mrnumber = {1821074}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1821/} }
TY - JOUR AU - Deninger, Christopher AU - Singhof, Wilhelm TI - A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas JO - Annales de l'Institut Fourier PY - 2001 SP - 209 EP - 219 VL - 51 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1821/ DO - 10.5802/aif.1821 LA - en ID - AIF_2001__51_1_209_0 ER -
%0 Journal Article %A Deninger, Christopher %A Singhof, Wilhelm %T A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas %J Annales de l'Institut Fourier %D 2001 %P 209-219 %V 51 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1821/ %R 10.5802/aif.1821 %G en %F AIF_2001__51_1_209_0
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, Volume 51 (2001) no. 1, pp. 209-219. doi : 10.5802/aif.1821. https://aif.centre-mersenne.org/articles/10.5802/aif.1821/
[AGH] Flows on homogeneous spaces, Ann. Math. Studies, 53, Princeton, 1963 | Zbl
[AK1] Long time behaviour of leafwise heat flow for Riemannian foliations (To appear in Compositio Math.) | Zbl
[AK2] Distributional Betti numbers of Lie Foliations (2000) (Manuscript)
[AT] Hodge decomposition along the leaves of a Riemannian foliation, J. Functional Analysis, Volume 99 (1991), pp. 443-458 | DOI | MR | Zbl
[CR] Global solvability on compact nilmanifolds, Transactions AMS, Volume 301 (1987), pp. 343-373 | DOI | MR | Zbl
[D1] 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] On dynamical systems and their possible significance for Arithmetic Geometry II (2000) (Preprint) | MR | Zbl
[DS] A note on dynamical trace formulas (2000) (Preprint) | MR | Zbl
[G] Lectures on spectral theory of elliptic operators, Duke Math. J., Volume 44 (1977), pp. 485-517 | DOI | MR | Zbl
[Go] Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc., Volume 143 (1969), pp. 55-76 | DOI | MR | Zbl
[GS] Geometric asymptotics, Math. Surveys, 14, Amer. Math. Soc., Providence, R.I, 1977 | MR | Zbl
[H] On Frobenius reciprocity for unipotent algebraic groups over , Amer. J. of Math., Volume 93 (1971), pp. 163-172 | DOI | MR | Zbl
[MS] Global Analysis on foliated spaces, MSRI Publications, 9, Springer, 1988 | MR | Zbl
[P] On Ruelle's zeta function, Israel Math. Conf. Proc., Volume 3 (1990), pp. 163-184 | MR | Zbl
[R1] Decomposition of the -space of a general compact nilmanifold, Amer. J. of Math., Volume 93 (1971), pp. 173-190 | DOI | MR | Zbl
[R2] Global solvability on compact Heisenberg manifolds, Transactions AMS (1982), pp. 309-317 | DOI | MR | Zbl
Cited by Sources: