With cedram.org
Table of contents for this issue | Previous article | Next article
Christopher Deninger; Wilhelm Singhof
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 traces dynami)
Annales de l'institut Fourier, 51 no. 1 (2001), p. 209-219, doi: 10.5802/aif.1821
Article PDF | Reviews MR 1821074 | Zbl 0997.58017
Class. Math.: 53C12, 37C27, 58J60
Keywords: Hodge decomposition, foliation, dynamical trace formula, nilmanifold

Résumé - Abstract

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.


[AK1] J.A. Álvarez López & Y. Kordyukov, “Long time behaviour of leafwise heat flow for Riemannian foliations”, To appear in Compositio Math. arXiv |  Zbl 0982.58019
[AK2] J.A. Álvarez López & Y. Kordyukov, “Distributional Betti numbers of Lie Foliations”, Manuscript, 2000
[AT] J.A. Álvarez López & P. Tondeur, “Hodge decomposition along the leaves of a Riemannian foliation”, J. Functional Analysis 99 (1991), p. 443-458 Article |  MR 1121621 |  Zbl 0746.58011
[AGH] L. Auslander, L. Green & F. Hahn, Flows on homogeneous spaces, Ann. Math. Studies 53, Princeton, 1963  Zbl 0106.36802
[CR] J.M. Cygan & L.F. Richardson, “Global solvability on compact nilmanifolds”, Transactions AMS 301 (1987), p. 343-373 Article |  MR 879578 |  Zbl 0631.58034
[D1] C. Deninger, Some analogies between number theory and dynamical systems on foliated spaces, Doc. Math. J. DMV, 1998, p. 23-46  Zbl 0899.14001
[D2] C. Deninger, “On dynamical systems and their possible significance for Arithmetic Geometry II”, Preprint, 2000  MR 1724887 |  Zbl 02106437
[DS] C. Deninger & W. Singhof, “A note on dynamical trace formulas”, Preprint, 2000  MR 1868467 |  Zbl 01787203
[Go] R. Goodman, “Analytic and entire vectors for representations of Lie groups”, Trans. Amer. Math. Soc. 143 (1969), p. 55-76 Article |  MR 248285 |  Zbl 0189.14102
[G] V. Guillemin, “Lectures on spectral theory of elliptic operators”, Duke Math. J. 44 (1977), p. 485-517 Article |  MR 448452 |  Zbl 0463.58024
[GS] V. Guillemin & S. Sternberg, Geometric asymptotics, Math. Surveys 14, Amer. Math. Soc., Providence, R.I, 1977  MR 516965 |  Zbl 0364.53011
[H] R. Howe, “On Frobenius reciprocity for unipotent algebraic groups over ${\scriptstyle\Q}$”, Amer. J. of Math. 93 (1971), p. 163-172 Article |  MR 281842 |  Zbl 0215.11803
[MS] C.C. Moore & C. Schochet, Global Analysis on foliated spaces, MSRI Publications 9, Springer, 1988  MR 918974 |  Zbl 0648.58034
[P] S.J. Patterson, “On Ruelle's zeta function”, Israel Math. Conf. Proc. 3 (1990), p. 163-184  MR 1159114 |  Zbl 0721.58041
[R1] L.F. Richardson, “Decomposition of the $L^2$-space of a general compact nilmanifold”, Amer. J. of Math. 93 (1971), p. 173-190 Article |  MR 284546 |  Zbl 0265.43012
[R2] L.F. Richardson, “Global solvability on compact Heisenberg manifolds”, Transactions AMS (1982), p. 309-317 Article |  MR 664044 |  Zbl 0503.35003