Vanishing of the entropy pseudonorm for certain integrable systems
Authors:
Boris S. Kruglikov and Vladimir S. Matveev
Journal:
Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 19-28
MSC (2000):
Primary 37C85, 37J35, 37B40; Secondary 70H07, 37A35
DOI:
https://doi.org/10.1090/S1079-6762-06-00156-9
Published electronically:
March 2, 2006
MathSciNet review:
2200951
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Abstract: We introduce the notion of entropy pseudonorm for an action of $\mathbb {R}^n$ and prove that it vanishes for the group actions associated with a large class of integrable Hamiltonian systems.
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B1 S. Benenti, Inertia tensors and Stäckel systems in the Euclidean spaces, Differential geometry (Turin, 1992). Rend. Sem. Mat. Univ. Politec. Torino 50 (1993), no. 4, 315–341.
B2 S. Benenti, An outline of the geometrical theory of the separation of variables in the Hamilton-Jacobi and Schrödinger equations, SPT 2002: Symmetry and perturbation theory (Cala Gonone), 10–17, World Sci. Publishing, River Edge, NJ, 2002.
B3 S. Benenti, Special Symmetric Two-tensors, Equivalent Dynamical Systems, Cofactor and Bi-Cofactor Systems, Acta Applicandae Mathematicae 87 (2005), no. 1-3, 33–91.
BF A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman & Hall, 2004.
BM A. V. Bolsinov, V. S. Matveev, Geometrical interpretation of Benenti’s systems, Journ. Geom. Phys. 44 (2003), 489–506.
BT A. V. Bolsinov, I. A. Taimanov, Integrable geodesic flows with positive topological entropy, Invent. Math. 140 (2000), 639–650.
Bo R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. A.M.S. 153 (1971), 401–414.
BP L. T. Butler, G. P. Paternain, Collective geodesic flows, Ann. Inst. Fourier (Grenoble) 53 (2003) no. 1, 265–308.
Bu L. T. Butler, Toda lattices and positive-entropy integrable systems, Invent. Math. 158 (2004), no. 3, 515–549.
CST M. Crampin, W. Sarlet, G. Thompson, Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors, J. Phys. A 33 (2000), no. 48, 8755–8770.
E L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals. Elliptic case, Comment. Math. Helv. 65 (1990), no. 1, 4–35.
C J. P. Conze, Entropie d’un groupe abelien de transformations, Z. Wahrsch. 25 (1972), 11–30.
GS V. Guillemin, S. Sternberg, On collective complete integrability according to the method of Thimm, Ergod. Th. & Dynam. Sys. 3 (1983), no. 2, 219–230.
HS K. H. Hofmann, L. N. Stojanov, Topological entropy of group and semigroup actions, Adv. Math. 115 (1995), no. 1, 54–98.
H Y. Hu, Some ergodic properties of commuting diffeomorphisms, Ergod. Th. & Dynam. Sys. 13 (1993), 73–100.
IMM A. Ibort, F. Magri, G. Marmo, Bihamiltonian structures and Stäckel separability, J. Geom. Phys. 33 (2000), no. 3-4, 210–228.
I H. Ito, Action-angle coordinates at singularities for analytic integrable systems, Math. Z. 206 (1991), 363–407.
KH A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Math. and its Appl. 54, Cambridge University Press, Cambridge, 1995.
K B. Kruglikov, Examples of integrable sub-Riemannian geodesic flows, Jour. Dynam. Contr. Syst. 8 (2002), no. 3, 323–340.
KM B. S. Kruglikov, V. S. Matveev, Strictly nonproportional geodesically equivalent metrics have $h_\textrm {top}(g)=0$, Ergod. Th. & Dynam. Sys. 26 (2006), 219–245.
LC T. Levi-Civita, Sulle trasformazioni delle equazioni dinamiche, Ann. di Mat., serie $2^a$, 24 (1896), 255–300.
LR H. Lundmark, S. Rauch-Wojciechowski, Driven Newton equations and separable time-dependent potentials, J. Math. Phys. 43 (2002), no. 12, 6166–6194.
MT V. Matveev, P. Topalov, Trajectory equivalence and corresponding integrals, Regular and Chaotic Dynamics 3 (1998), no. 2, 30–45.
M1 V. S. Matveev, Hyperbolic manifolds are geodesically rigid, Invent. Math. 151 (2003), 579–609.
M2 V. S. Matveev, Three-dimensional manifolds having metrics with the same geodesics, Topology 42 (2003), no. 6, 1371–1395.
M3 V. S. Matveev, Projectively equivalent metrics on the torus, Diff. Geom. Appl. 20 (2004), 251–265.
P1 G. Paternain, On the topology of manifolds with completely integrable geodesic flows. I, Ergod. Th. & Dynam. Sys. 12 (1992), 109–121.
P2 G. Paternain, On the topology of manifolds with completely integrable geodesic flows. II, Journ. Geom. Phys. 123 (1994), 289–298.
P3 G. Paternain, Geodesic flows, Birkhäuser, 1999.
PP G. P. Paternain, J. Petean, Zero entropy and bounded topology, preprint: ArXiv.org/math.DG/0406051; to appear in Comment. Math. Helv.
Pe Y. Pesin, Dimension theory in dynamical systems, Chicago Lect. in Math. Ser., The University of Chicago Press, 1997.
T I. A. Taimanov, Topology of Riemannian manifolds with integrable geodesic flows, Proc. Steklov Inst. Math. 205 (1995), 139–150.
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Additional Information
Boris S. Kruglikov
Affiliation:
Institute of Mathematics and Statistics, University of Tromsø, Tromsø90-37, Norway
Email:
kruglikov@math.uit.no
Vladimir S. Matveev
Affiliation:
Mathematisches Institut der Albert-Ludwigs-Universität, Eckerstraße-1, Freiburg 79104, Germany
MR Author ID:
609466
Email:
matveev@email.mathematik.uni-freiburg.de
Received by editor(s):
October 4, 2005
Published electronically:
March 2, 2006
Communicated by:
Boris Hasselblatt
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
