Abstract
The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence. This also establishes a connection between Floer homology and geometry of 3-manifolds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arnoux P, Yoccoz J.-C: Construction de difféomorphismes pseudo-Anosov. C. R. Acad. Sci. Paris Sér. I Math. 292, 75–78 (1981)
Artin M., Mazur B: On periodic points. Ann. of Math. (2)81, 82–99 (1965)
Brock J.F: Weil-Petersson translation distance and volumes of mapping tori. Comm. Anal. Geom. 11: 987–999 (2003)
Cotton-Clay A.: Symplectic Floer homology of area-preserving surface diffeomorphisms. Geom. Topol. 13, 2619–2674 (2009)
A. Cotton-Clay, A sharp bound on fixed points of surface symplectomorphisms in each mapping class. arXiv: 1009.0760[math.SG], 2010.
Dostoglou S., D. Salamon: Self-dual instantons and holomorphic curves. Ann. of Math. (2) 139, 581– (1994)
A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les Surfaces. Astérisque 66, Soc. Math. France, 1979.
A. L. Fel’shtyn, New zeta function in dynamic. In: Tenth Internat. Conf. on Nonlinear Oscillations, Abstracts of Papers, Varna, Bulgar. Acad. Sci., 1984.
A. L. Fel’shtyn, New zeta functions for dynamical systems and Nielsen fixed point theory. In: Topology and Geometry—Rohlin Seminar, Lecture Notes in Math. 1346, Springer, 1988, 33–55.
A. Fel’shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion. Mem. Amer. Math. Soc. 147 (2000), xii+146.
A. L. Fel’shtyn, Floer homology, Nielsen theory and symplectic zeta functions. In: Proceedings of the Steklov Institute of Mathematics, Moscow, vol. 246, 2004, 270–282.
A. L. Fel’shtyn: Nielsen theory, Floer homology and a generalisation of the Poincare-Birkhoff theorem. J. Fixed Point Theory Appl. 3, 191–214 (2008)
A. L. Fel’shtyn and R. Hill, The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion. K-Theory 8 (1994), 367–393.
A. L. Fel’shtyn, R. Hill: Trace formulae, zeta functions, congruences and Reidemeister torsion in Nielsen theory. Forum Math. 10, 641–663 (1998)
A. Floer, Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), 513–547.
R. Fintushel, R. Stern: Knots, links and 4-manifolds. Invent. Math. 134, 363–400 (1998)
M. Handel, The entropy of orientation reversing homeomorphisms of surfaces. Topology 21 (1982), 291–296.
E. Hironaka, E. Kin: A family of pseudo-Anosov braids with small dilatation. Algebr. Geom. Topol. 6, 699–738 (2006)
R. Gautschi, Floer homology of algebraically finite mapping classes. J. Symplectic Geom. 1 (2003), 715–765.
M. Gromov: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)
E.-N. Ionel and Th. Parker, Gromov invariants and symplectic maps. Math. Ann. 314 (1999), 127–158.
N. V. Ivanov, Entropy and the Nielsen numbers. Dokl. Akad. Nauk SSSR 265 (1982), 284–287 (in Russian); English transl.: Soviet Math. Dokl. 26 (1982), 63–66.
N. V. Ivanov: Coefficients of expansion of pseudo-Anosov homeomorphisms. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167, 111– (1988)
B. Jiang, Estimation of the number of periodic orbits. Pacific J. Math. 172 (1996), 151–185.
T. Kobayashi, Links of homeomorphisms of surfaces and topological entropy. Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), 381–383.
M. Linch: A comparison of metrics on Teichmüller space. Proc. Amer. Math. Soc. 43, 349–352 (1974)
Manning A: Axiom A diffeomorphisms have rational zeta function. Bull. Lond. Math. Soc. 3, 215–220 (1971)
D. McDuff and D. A. Salamon, J-Holomorphic Curves and Symplectic Topology. Amer. Math. Soc. Colloq. Publ. 52, Amer. Math. Soc., Providence, RI, 2004.
C. T. McMullen: Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations. Ann. Sci. école Norm. Sup. (4) 33, 519–560 (2000)
H. Minakawa, Examples of pseudo-Anosov homeomorphisms with small dilatations. J. Math. Sci. Univ. Tokyo 13 (2006), 95–111.
J. Moser, On the volume elements on a manifold. Trans. Amer. Math. Soc. 120 (1965), 286–294.
R. C. Penner: Bounds on least dilatations. Proc. Amer. Math. Soc. 113 (1991), 443–450.
V. B. Pilyugina and A. L. Fel’shtyn, The Nielsen zeta function. Funktsional. Anal. i Prilozhen. 19 (1985), 61–67 (in Russian); English transl.: Functional Anal. Appl. 19 (1985), 300–305.
M. Pozniak, Floer homology, Novikov rings and clean intersections. PhD thesis, University of Warwick, 1994.
P. Seidel: Symplectic Floer homology and the mapping class group. Pacific J. Math. 206, 219–229 (2002)
P. Seidel, Braids and symplectic four-manifolds with abelian fundamental group. Turkish J. Math. 26 (2002), 93–100.
M. Shub, Dynamical systems, filtrations and entropy. Bull. Amer. Math. Soc. 80 (1974), 27–41.
I. Smith, Floer cohomology and pencils of quadrics. arXiv: 1006. 1099v1[math.SG], 2010.
D. Sullivan, Travaux de Thurston sur les groupes quasi-Fuchsiens et les varietes hyperboliques de dimension 3 fibres sur S 1. In: Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math. 842, Springer, Berlin, 1981, 1–19.
W. P. Thurston: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417–431.
W. Thurston, Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fibers over the circle. Preprint, arXiv: math/9801045[math.GT].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Fel’shtyn, A. The growth rate of symplectic Floer homology. J. Fixed Point Theory Appl. 12, 93–119 (2012). https://doi.org/10.1007/s11784-013-0098-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-013-0098-3