Abstract
Let T n denote the group of real n × n upper-triangular matrices with 1s on the diagonal. This paper constructs left-invariant Riemannian and sub-Riemannian metrics on T 3 ⊕ T 4 whose geodesic flow has a subsystem that factors onto a suspended horseshoe. As a corollary, left-invariant Riemannian metrics with positive topological entropy are constructed on all quotients D∖T n where D is a discrete subgroup of T n and n ≥ 7.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bowen, R.: Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414.
Butler, L. T. and Paternain,G. P.: Collective geodesic flows, Ann. Inst. Fourier. to appear.
Butler, L. T.: Integrable geodesic flows on n-step nilmanifolds. J. Geom. Phys. 36(3–4) (2000), 315–323.
Butler, L. T.: Zero entropy, nonintegrable geodesic flows and a non-commutative rotation vector, Trans. Amer. Math. Soc. 355 (2003), 3641–3650.
Butler, L. T.: Integrable geodesic flows with wild first integrals: the case of two-step nilmanifolds, Ergodic Theory Dynam. Systems, 23(3) (2003), 777–797.
Delshams, A. and Seara, T. M.: An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys. 150(3) (1992), 433–463.
Guillemin, V. and Sternberg, S.: Symplectic Techniques in Physics, 2nd edn, Cambridge University Press, Cambridge, 1990.
Karidi, R.: Geometry of balls in nilpotent Lie groups, Duke Math. J. 74 (1994), 301–317.
Holmes, P. and Marsden, J.: Horseshoes and Arnol'd diffusion for Hamiltonian systems on Lie groups, Indiana Univ. Math. J. 32 (1983), 273–309.
Mañé, R.: On the topological entropy of geodesic flows, J. Differential Geom. 45 (1997), 74–93.
Manning, A.: More topological entropy for geodesic flows, In: Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math. 898, Springer, New York, 1980, pp. 243–249.
Marsden, J. E. and Ratiu, T. S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd edn, Texts in Appl. Math. 17, Springer, New York, 1999.
Montgomery, R., Shapiro, M. and Stolin, A.: A nonintegrable sub-Riemannian geodesic flow on a Carnot group, J. Dynam. Control Systems 3(4) (1998), 519–530.
Paternain, G. P.: Geodesic Flows, Birkhäuser, Basel, 1999.
Robinson, C.: Horseshoes for autonomous Hamiltonian systems using the Melnikov integral, Ergodic Theory Dynam. Systems, 8* (Charles Conley Memorial Issue) (1988), 395–409.
Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd edn, CRC Press, Boca Raton, FL, 1999.
Ziglin, S. L.: Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics., I, Funktsional. Anal. i Prilozhen. 16(3) (1982), 30–41.
Ziglin, S. L.: Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics., II, Funktsional. Anal. i Prilozhen. 17(1) (1983), 8–23.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Butler, L.T. Invariant Metrics on Nilmanifolds with Positive Topological Entropy. Geometriae Dedicata 100, 173–185 (2003). https://doi.org/10.1023/A:1025838402716
Issue Date:
DOI: https://doi.org/10.1023/A:1025838402716