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Positive topological entropy of Reeb flows on spherizations



Let M be a closed manifold whose based loop space Ω (M) is “complicated”. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T*M which is fiberwise starshaped with respect to the origin. Choose a function H : T*M → ℝ such that Σ is a regular energy surface of H, and let ϕt be the restriction to Σ of the Hamiltonian flow of H.

Theorem 1. The topological entropy of ϕt is positive.

This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain–Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [19], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows.

Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T*M is positive.

For qM abbreviate Σq = Σ ∩ Tq*M. The following corollary extends results of Morse and Gromov on the number of geodesics between two points.

Corollary 1. Given q ∈ M, for almost every q′ ∈ M the number of orbits of the flow ϕt from Σq to Σq′ grows exponentially in time.

In the lowest dimension, Theorem 1 yields the existence of many closed, orbits.

Corollary 2. Let M be a closed surface different from S2, ℝP2, the torus and the Klein bottle. Then ϕt carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.



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[1]Abbondandolo, A. and Majer, P. Lectures on the Morse complex for infinite-dimensional manifolds. Morse theoretic methods in nonlinear analysis and in symplectic topology, 174. NATO Sci. Ser. II Math. Phys. Chem. 217 (Springer, 2006).
[2]Abbondandolo, A. and Schwarz, M.On the Floer homology of cotangent bundles. Comm. Pure Appl. Math. 59 (2006), 254316.
[3]Benci, V.Periodic solutions of Lagrangian systems on a compact manifold. J. Diff. Eq. 63 (1986), 135161.
[4]Biran, P., Polterovich, L. and Salamon, D.Propagation in Hamiltonian dynamics and relative symplectic homology. Duke Math. J. 119 (2003), 65118.
[5]Bowen, R.Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401414.
[6]Butler, L. and Paternain, G.Magnetic flows on Sol-manifolds: dynamical and symplectic aspects. Comm. Math. Phys. 284 (2008), 187202.
[7]Cieliebak, K., Floer, A. and Hofer, H.Symplectic homology. II. A general construction. Math. Z. 218 (1995), 103122.
[8]Colin, V. and Honda, K. Reeb vector fields and open book decompositions. arXiv:0809.5088.
[9]Dinaburg, E. I.A connection between various entropy characterizations of dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 324366.
[10]Eliashberg, Y., Kim, S.S. and Polterovich, L.Geometry of contact transformations and domains: orderability versus squeezing. Geom. Topol. 10 (2006), 16351747.
[11]Floer, A.A relative Morse index for the symplectic action. Comm. Pure Appl. Math. 41 (1988), 393407.
[12]Floer, A.The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math. 41 (1988), 775813.
[13]Floer, A.Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), 513547.
[14]Floer, A.Witten's complex and infinite-dimensional Morse theory. J. Differential Geom. 30 (1989), 207221.
[15]Floer, A.Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. 120 (1989), 575611.
[16]Floer, A. and Hofer, H.Symplectic homology. I. Open sets in Cn. Math. Z. 215 (1994), 3788.
[17]Floer, A., Hofer, H. and Salamon, D.Transversality in elliptic Morse theory for the symplectic action. Duke Math. J. 80 (1995), 251292.
[18]Frauenfelder, U. and Schlenk, F.Volume growth in the component of the Dehn–Seidel twist. Geom. Funct. Anal. 15 (2005), 809838.
[19]Frauenfelder, U. and Schlenk, F.Fiberwise volume growth via Lagrangian intersections. J. Symplectic Geom. 4 (2006), 117148.
[20]Frauenfelder, U. and Schlenk, F.Hamiltonian dynamics on convex symplectic manifolds. Israel J. Math. 159 (2007), 156.
[21]Gromov, M.Homotopical effects of dilatation. J. Differential Geom. 13 (1978), 303310.
[22]Heistercamp, M. The Weinstein conjecture with multiplicities on spherizations. PhD Thesis Université de Neuchâtel. In preparation.
[23]Heistercamp, M., Macarini, L. and Schlenk, F. Energy surfaces in ℝ2n and in cotangent bundles – convex versus starshaped. In preparation.
[24]Katok, A.Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137173.
[25]Katok, A.Entropy and closed geodesics. Ergodic Theory Dynam. Systems 2 (1982), 339365 (1983)
[26]Katok, A. and Hasselblatt, B.Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications 54 (Cambridge University Press, 1995).
[27]Khovanov, M. and Seidel, P.Quivers, Floer cohomology, and braid group actions. J. Amer. Math. Soc. 15 (2002), 203271.
[28]Klingenberg, W. Riemannian geometry. Second edition. de Gruyter Studies in Mathematics, 1 (de Gruyter, 1995).
[29]Macarini, L., Merry, W. and Paternain, G. On the growth rate of leaf-wise intersections, arXiv: 1101.4812.
[30]Macarini, L. and Paternain, G.On the stability of Mañé critical hypersurfaces. Calc. Var. Partial Differential Equations 39 (2010), 579591.
[31]Mañé, R.On the topological entropy of geodesic flows. J. Differential Geom. 45 (1997), 7493.
[32]Milnor, J.A note on curvature and fundamental group. J. Differential Geom. 2 (1968), 17.
[33]Newhouse, S.Entropy and volume. Ergodic Theory Dynam. Systems 8* (1988), Charles Conley Memorial Issue, 283299.
[34]Niche, C.Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete Contin. Dyn. Syst. 11 (2004), 577580.
[35]Paternain, G.Topological entropy for geodesic flows on fibre bundles over rationally hyperbolic manifolds. Proc. Amer. Math. Soc. 125 (1997), 27592765.
[36]Paternain, G. Geodesic flows. Progr. Math. 180 (Birkhäuser Boston, Inc., 1999).
[37]Paternain, G. and Paternain, M.Topological entropy versus geodesic entropy. Internat. J. Math. 5 (1994), 213218.
[38]Paternain, G. and Petean, J.Zero entropy and bounded topology. Comment. Math. Helv. 81 (2006), 287304.
[39]Salamon, D.Morse theory, the Conley index and Floer homology. Bull. London Math. Soc. 22 (1990), 113140.
[40]Salamon, D. and Zehnder, E.Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Comm. Pure Appl. Math. 45 (1992), 13031360.
[41]Schwarz, M. Morse homology. Progr. Math. 111 (Birkhäuser Verlag, 1993).
[42]Seidel, P.A biased view of symplectic cohomology. Current Developments in Mathematics 2006 (2008), 211253.
[43]Viterbo, C.Functors and computations in Floer homology with applications. I. Geom. Funct. Anal. 9 (1999), 9851033.
[44]Wullschleger, R. Slow entropy of Reeb flows in spherizations. PhD Thesis Université de Neuchâtel. In preparation.
[45]Yomdin, Y.Volume growth and entropy. Israel J. Math. 57 (1987), 285300.

Positive topological entropy of Reeb flows on spherizations



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