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Vanishing of the entropy pseudonorm for certain integrable systems
Author(s):
Boris
S.
Kruglikov;
Vladimir
S.
Matveev
Journal:
Electron. Res. Announc. Amer. Math. Soc.
12
(2006),
19-28.
MSC (2000):
Primary 37C85, 37J35, 37B40;
Secondary 70H07, 37A35
Posted:
March 2, 2006
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Additional information
Abstract:
We introduce the notion of entropy pseudonorm for an action of and prove that it vanishes for the group actions associated with a large class of integrable Hamiltonian systems.
References:
-
- 1.
- V.I. Arnold, Mathematical methods of classical mechanics, Nauka, Moscow; English transl., Graduate Texts in Mathematics, Springer, 1989. MR 1037020 (93c:70001)
- 2.
- 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. MR 1261446 (95c:70020)
- 3.
- 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. MR 1976651 (2004a:70022)
- 4.
- S. Benenti, Special Symmetric Two-tensors, Equivalent Dynamical Systems, Cofactor and Bi-Cofactor Systems, Acta Applicandae Mathematicae 87 (2005), no. 1-3, 33-91. MR 2151124
- 5.
- A.V. Bolsinov, A.T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman & Hall, 2004. MR 2036760 (2004j:37106)
- 6.
- A.V. Bolsinov, V.S. Matveev, Geometrical interpretation of Benenti's systems, Journ. Geom. Phys. 44 (2003), 489-506. MR 1943174 (2005d:37117)
- 7.
- A.V. Bolsinov, I.A. Taimanov, Integrable geodesic flows with positive topological entropy, Invent. Math. 140 (2000), 639-650. MR 1760753 (2001b:37081)
- 8.
- R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. A.M.S. 153 (1971), 401-414. MR 0274707 (43:469)
- 9.
- L.T. Butler, G.P. Paternain, Collective geodesic flows, Ann. Inst. Fourier (Grenoble) 53 (2003) no. 1, 265-308. MR 1973073 (2004m:37054)
- 10.
- L.T. Butler, Toda lattices and positive-entropy integrable systems, Invent. Math. 158 (2004), no. 3, 515-549. MR 2104793 (2005h:37122)
- 11.
- 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. MR 1801467 (2001k:37094)
- 12.
- L.H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals. Elliptic case, Comment. Math. Helv. 65 (1990), no. 1, 4-35. MR 1036125 (91d:58223)
- 13.
- J.P. Conze, Entropie d'un groupe abelien de transformations, Z. Wahrsch. 25 (1972), 11-30. MR 0335754 (49:534)
- 14.
- V. Guillemin, S. Sternberg, On collective complete integrability according to the method of Thimm, Ergod. Th. & Dynam. Sys. 3 (1983), no. 2, 219-230. MR 0742224 (85i:58057)
- 15.
- K.H. Hofmann, L.N. Stojanov, Topological entropy of group and semigroup actions, Adv. Math. 115 (1995), no. 1, 54-98. MR 1351326 (97a:22006)
- 16.
- Y. Hu, Some ergodic properties of commuting diffeomorphisms, Ergod. Th. & Dynam. Sys. 13 (1993), 73-100. MR 1213080 (94b:58061)
- 17.
- A. Ibort, F. Magri, G. Marmo, Bihamiltonian structures and Stäckel separability, J. Geom. Phys. 33 (2000), no. 3-4, 210-228. MR 1747040 (2001a:37082)
- 18.
- H. Ito, Action-angle coordinates at singularities for analytic integrable systems, Math. Z. 206 (1991), 363-407. MR 1095762 (92e:58069)
- 19.
- A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Math. and its Appl. 54, Cambridge University Press, Cambridge, 1995. MR 1326374 (96c:58055)
- 20.
- B. Kruglikov, Examples of integrable sub-Riemannian geodesic flows, Jour. Dynam. Contr. Syst. 8 (2002), no. 3, 323-340. MR 1914446 (2003m:37081)
- 21.
- B. S. Kruglikov, V. S. Matveev, Strictly nonproportional geodesically equivalent metrics have
, Ergod. Th. & Dynam. Sys. 26 (2006), 219-245. - 22.
- T. Levi-Civita, Sulle trasformazioni delle equazioni dinamiche, Ann. di Mat., serie
, 24 (1896), 255-300. - 23.
- H. Lundmark, S. Rauch-Wojciechowski, Driven Newton equations and separable time-dependent potentials, J. Math. Phys. 43 (2002), no. 12, 6166-6194. MR 1939638 (2004j:70035)
- 24.
- V. Matveev, P. Topalov, Trajectory equivalence and corresponding integrals, Regular and Chaotic Dynamics 3 (1998), no. 2, 30-45. MR 1693470 (2000d:37068)
- 25.
- V.S. Matveev, Hyperbolic manifolds are geodesically rigid, Invent. Math. 151 (2003), 579-609. MR 1961339 (2004f:53044)
- 26.
- V.S. Matveev, Three-dimensional manifolds having metrics with the same geodesics, Topology 42 (2003), no. 6, 1371-1395. MR 1981360 (2004g:53095)
- 27.
- V.S. Matveev, Projectively equivalent metrics on the torus, Diff. Geom. Appl. 20 (2004), 251-265. MR 2053913 (2005c:53108)
- 28.
- G. Paternain, On the topology of manifolds with completely integrable geodesic flows. I, Ergod. Th. & Dynam. Sys. 12 (1992), 109-121. MR 1162403 (93g:58117)
- 29.
- G. Paternain, On the topology of manifolds with completely integrable geodesic flows. II, Journ. Geom. Phys. 123 (1994), 289-298. MR 1269245 (95b:58115)
- 30.
- G. Paternain, Geodesic flows, Birkhäuser, 1999. MR 1712465 (2000h:53108)
- 31.
- G.P. Paternain, J. Petean, Zero entropy and bounded topology, preprint: ArXiv.org/math.DG/0406051; to appear in Comment. Math. Helv.
- 32.
- Y. Pesin, Dimension theory in dynamical systems, Chicago Lect. in Math. Ser., The University of Chicago Press, 1997. MR 1489237 (99b:58003)
- 33.
- I.A. Taimanov, Topology of Riemannian manifolds with integrable geodesic flows, Proc. Steklov Inst. Math. 205 (1995), 139-150. MR 1428676 (98b:58135)
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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
Email:
matveev@email.mathematik.uni-freiburg.de
DOI:
10.1090/S1079-6762-06-00156-9
PII:
S 1079-6762(06)00156-9
Received by editor(s):
October 4, 2005
Posted:
March 2, 2006
Communicated by:
Boris Hasselblatt
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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