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A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas

Christopher Deninger[1]; Wilhelm Singhof[2]

  • [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 (2001)

  • Volume: 51, Issue: 1, page 209-219
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Deninger, Christopher, and 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 51.1 (2001): 209-219. <http://eudml.org/doc/115909>.

@article{Deninger2001,
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.},
affiliation = {WWU Münster, Mathematisches Institut, Einsteinstr. 62, 48149 Münster (Allemagne); Universität Düsseldorf, Mathematisches Institut, Universitätsstr. 1, 40225 Düsseldorf (Allemagne)},
author = {Deninger, Christopher, Singhof, Wilhelm},
journal = {Annales de l’institut Fourier},
keywords = {Hodge decomposition; foliation; dynamical trace formula; nilmanifold},
language = {eng},
number = {1},
pages = {209-219},
publisher = {Association des Annales de l'Institut Fourier},
title = {A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas},
url = {http://eudml.org/doc/115909},
volume = {51},
year = {2001},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 1
SP - 209
EP - 219
AB - 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.
LA - eng
KW - Hodge decomposition; foliation; dynamical trace formula; nilmanifold
UR - http://eudml.org/doc/115909
ER -

References

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  2. J.A. Álvarez López, Y. Kordyukov, Distributional Betti numbers of Lie Foliations, (2000) 
  3. J.A. Álvarez López, P. Tondeur, Hodge decomposition along the leaves of a Riemannian foliation, J. Functional Analysis 99 (1991), 443-458 Zbl0746.58011MR1121621
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  6. C. Deninger, Some analogies between number theory and dynamical systems on foliated spaces, ICM Extra Volume I (1998), 23-46 Zbl0899.14001
  7. C. Deninger, On dynamical systems and their possible significance for Arithmetic Geometry II, (2000) Zbl1159.11310MR1724887
  8. C. Deninger, W. Singhof, A note on dynamical trace formulas, (2000) Zbl1211.58016MR1868467
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  11. V. Guillemin, S. Sternberg, Geometric asymptotics, 14 (1977), Amer. Math. Soc., Providence, R.I Zbl0364.53011MR516965
  12. R. Howe, On Frobenius reciprocity for unipotent algebraic groups over ℚ , Amer. J. of Math. 93 (1971), 163-172 Zbl0215.11803MR281842
  13. C.C. Moore, C. Schochet, Global Analysis on foliated spaces, 9 (1988), Springer Zbl0648.58034MR918974
  14. S.J. Patterson, On Ruelle's zeta function, Israel Math. Conf. Proc. 3 (1990), 163-184 Zbl0721.58041MR1159114
  15. L.F. Richardson, Decomposition of the L 2 -space of a general compact nilmanifold, Amer. J. of Math. 93 (1971), 173-190 Zbl0265.43012MR284546
  16. L.F. Richardson, Global solvability on compact Heisenberg manifolds, Transactions AMS (1982), 309-317 Zbl0503.35003MR664044

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