Advertisement

Israel Journal of Mathematics

, Volume 224, Issue 1, pp 367–383 | Cite as

Lower complexity bounds for positive contactomorphisms

  • Lucas Dahinden
Article
  • 31 Downloads

Abstract

Let S*Q be the spherization of a closed connected manifold of dimension at least two. Consider a contactomorphism φ that can be reached by a contact isotopy that is everywhere positively transverse to the contact structure. In other words, φ is the time-1-map of a time-dependent Reeb flow. We show that the volume growth of φ is bounded from below by the topological complexity of the loop space of Q. Denote by ΩQ0(q) the component of the based loop space that contains the constant loop.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Albers and U. Fauenfelder, A variational approach to Givental’s nonlinear Maslov index, Geometric and Functional Analysis 22 (2012), 1033–1050.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. R. R. Alves, Legendrian contact homology and topological entropy, arXiv:1410.3381.Google Scholar
  3. [3]
    M. R. R. Alves, Cylindrical contact homology and topological entropy, Geometry & Topology 20 (2016), 3519–3569.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. R. R. Alves, Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds, Journal of Modern Dynamics 10 (2016), 497–509.MathSciNetCrossRefGoogle Scholar
  5. [5]
    K. Cieliebak and U. A. Frauenfelder, A Floer homology for exact contact embeddings, Pacific Journal of Mathematics 239 (2009), 251–316.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 35 (1971), 324–366.MathSciNetGoogle Scholar
  7. [7]
    U. Frauenfelder and F. Schlenk, Volume growth in the component of the Dehn–Seidel twist, Geometric and Functional Analysis 15 (2005), 809–838.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    U. Frauenfelder, C. Labrousse and F. Schlenk, Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems, Journal of Topology and Analysis 7 (2015), 407–451.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    U. Frauenfelder and F. Schlenk, Fiberwise volume growth via Lagrangian intersections, The Journal of Symplectic Geometry 4 (2006), 117–148.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    U. Frauenfelder and F. Schlenk, Filtered Hopf algebras and counting geodesic chords, Mathematische Annalen 360 (2014), 995–1020.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. Gromov, Homotopical effects of dilatation, Journal of Differential Geometry 13 (1978), 303–310.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, Vol. 54, Cambridge University Press, Cambridge, 1995.Google Scholar
  13. [13]
    L. Macarini and F. Schlenk, Positive topological entropy of Reeb flows on spherizations, Mathematical Proceedings of the Cambridge Philosophical Society 151 (2011), 103–128.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    W. J. Merry, Lagrangian Rabinowitz Floer homology and twisted cotangent bundles, Geometriae Dedicata 171 (2014), 345–386.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S. E. Newhouse, Entropy and volume, Ergodic Theory and Dynamical Systems 8* (1988), 283–299.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    G. P. Paternain, Topological entropy for geodesic flows on fibre bundles over rationally hyperbolic manifolds, Proceedings of the American Mathematical Society 125 (1997), 2759–2765.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    G. P. Paternain, Geodesic Flows, Progress in Mathematics, Vol. 180, Birkhäuser Boston, Inc., Boston, MA, 1999.Google Scholar
  18. [18]
    G. P. Paternain and J. Petean, Minimal entropy and collapsing with curvature bounded from below, Inventiones Mathematicae 151 (2003), 415–450.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    G. P. Paternain and J. Petean, Zero entropy and bounded topology, Commentarii Mathematici Helvetici 81 (2006), 287–304.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Y. Yomdin, Volume growth and entropy, Israel Journal of Mathematics 57 (1987), 285–300.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité de Neuchâtel (UNINE)NeuchâtelSwitzerland

Personalised recommendations