Geodesic
On Lefschetz formulas for flows on foliated manifolds
Ufa mathematical journal, Tome 7 (2015) no. 2, pp. 71-101 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper is devoted to the Lefschetz formulas for flows on compact manifolds, preserving a codimension one foliation and having fixed points. We develop an approach to the Lefschetz formulae based on the notion of the regularized trace on some algebra of singular integral operators introduced in a previous paper. The Lefschetz formula is proved in the case when the flow preserves a foliation given by the fibers of a fiber bundle over a circle. For a particular example of a flow on a two-dimensional torus, preserving a Reeb type foliation, we prove an analogue of the McKean–Singer formula for smoothed regularized Lefschetz functions.
Keywords: flow, closed orbits, fixed points, foliated manifold, regularized trace.
Mots-clés : Lefschetz formula 
@article{UFA_2015_7_2_a5,
     author = {Y. A. Kordyukov and V. A. Pavlenko},
     title = {On {Lefschetz} formulas for flows on foliated manifolds},
     journal = {Ufa mathematical journal},
     pages = {71--101},
     year = {2015},
     volume = {7},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2015_7_2_a5/}
}
TY  - JOUR
AU  - Y. A. Kordyukov
AU  - V. A. Pavlenko
TI  - On Lefschetz formulas for flows on foliated manifolds
JO  - Ufa mathematical journal
PY  - 2015
SP  - 71
EP  - 101
VL  - 7
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/UFA_2015_7_2_a5/
LA  - en
ID  - UFA_2015_7_2_a5
ER  - 
%0 Journal Article
%A Y. A. Kordyukov
%A V. A. Pavlenko
%T On Lefschetz formulas for flows on foliated manifolds
%J Ufa mathematical journal
%D 2015
%P 71-101
%V 7
%N 2
%U http://geodesic.mathdoc.fr/item/UFA_2015_7_2_a5/
%G en
%F UFA_2015_7_2_a5
Y. A. Kordyukov; V. A. Pavlenko. On Lefschetz formulas for flows on foliated manifolds. Ufa mathematical journal, Tome 7 (2015) no. 2, pp. 71-101. http://geodesic.mathdoc.fr/item/UFA_2015_7_2_a5/

[1] J. Álvarez López, Yu. A. Kordyukov, “Distributional Betti numbers of transitive foliations of codimension one”, Foliations: Geometry and Dynamics (Warsaw, 2000), World Sci. Publishing, River Edge, NJ, 2002, 159–183 | MR | Zbl

[2] A. Deitmar, “A higher rank Lefschetz formula”, J. Fixed Point Theory Appl., 2:1 (2007), 1–40 | DOI | MR | Zbl

[3] Ch. Deninger, “Some analogies between number theory and dynamical systems on foliated spaces”, Proceedings of the International Congress of Mathematicians (Berlin, 1998), v. I, Doc. Math., Extra Vol. I, 1998, 163–186, (electronic) | MR | Zbl

[4] Ch. Deninger, “Number theory and dynamical systems on foliated spaces”, Jahresber. Deutsch. Math.-Verein., 103:3 (2001), 79–100 | MR | Zbl

[5] Ch. Deninger, “Analogies between analysis on foliated spaces and arithmetic geometry”, Groups and analysis, London Math. Soc. Lecture Note Ser., 354, Cambridge Univ. Press, Cambridge, 2008, 174–190 | MR | Zbl

[6] D. Fried, “Lefschetz formulas for flows”, The Lefschetz centennial conference, Part III (Mexico City, 1984), Contemp. Math., 58, Amer. Math. Soc., Providence, RI, 1987, 19–69 | DOI | MR

[7] A. Juhl, Cohomological theory of dynamical zeta functions, Progress in Mathematics, 194, Birkhauser Verlag, Basel, 2001 | MR | Zbl

[8] Yu. A. Kordyukov, “Functional calculus for tangentially elliptic operators on foliated manifolds”, Analysis and Geometry in Foliated Manifolds (Santiago de Compostela, 1994), World Sci. Publishing, River Edge, NJ, 1995, 113–136 | MR | Zbl

[9] Kordyukov Yu. A., “Teoriya indeksa i nekommutativnaya geometriya na mnogoobraziyakh so sloeniem”, UMN, 64:2(386) (2009), 73–202 | DOI | MR | Zbl

[10] Kordyukov Yu. A., Pavlenko V. A., “Singulyarnye integralnye operatory na mnogoobrazii s otmechennym podmnogoobraziem”, Ufimsk. matem. zhurn., 6:3 (2014), 35–71

[11] E. Leichtnam, “An invitation to Deninger's work on arithmetic zeta functions”, Geometry, spectral theory, groups, and dynamics, Contemp. Math., 387, Amer. Math. Soc., Providence, RI, 2005, 201–236 | DOI | MR | Zbl

[12] R. B. Melrose, “Pseudodifferential operators, corners and singular limits”, Proceedings of the International Congress of Mathematicians (Kyoto, 1990), v. I, II, Math. Soc. Japan, Tokyo, 1991, 217–234 | MR | Zbl

[13] R. B. Melrose, “Calculus of conormal distributions on manifolds with corners”, Internat. Math. Res. Notices, 1992:3 (1992), 51–61 | DOI | MR | Zbl

[14] R. B. Melrose, The Atiyah–Patodi–Singer index theorem, Research Notes in Mathematics, 4, A K Peters, Ltd., Wellesley, MA, 1993 | MR | Zbl

[15] J. Roe, “Finite propagation speed and Connes' foliation algebra”, Math. Proc. Cambridge Philos. Soc., 102 (1987), 459–466 | DOI | MR | Zbl