Area-preserving diffeomorphisms from the \(C^{1}\) standpoint. (English) Zbl 1259.37023
Peixoto, Maurício Matos (ed.) et al., Dynamics, games and science II. DYNA 2008, in honor of Maurício Peixoto and David Rand, University of Minho, Braga, Portugal, September 8–12, 2008. Papers based on talks given at the international conference. Berlin: Springer (ISBN 978-3-642-14787-6/hbk; 978-3-642-14788-3/ebook). Springer Proceedings in Mathematics 2, 191-222 (2011).
Summary: More than thirty years have passed since S. E. Newhouse [Am. J. Math. 99, 1061–1087 (1977; Zbl 0379.58011)] published a dichotomy on \(C^{1}\) area-preserving diffeomorphisms. Here we revisit some central results on surface conservative \(C^{1}\)-diffeomorphisms by presenting, in particular, a new proof of Newhouse’s theorem and also by proving some, although folklore, not yet proved results on this setting. We intend that this exposition can be used by a large audience as an introduction to the concept of dominated splitting and its relevance to the theory of \(C^{1}\)-stability of area-preserving diffeomorphisms.
For the entire collection see [Zbl 1215.00024].
For the entire collection see [Zbl 1215.00024].
MSC:
| 37D30 | Partially hyperbolic systems and dominated splittings |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37C35 | Orbit growth in dynamical systems |
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
Citations:
Zbl 0379.58011References:
| [1] | Araújo, V.; Bessa, M., Dominated splitting, singularities and zero volume for incompressible 3-flows, Nonlinearity, 21, 1637-1653 (2008) · Zbl 1223.37036 · doi:10.1088/0951-7715/21/7/014 |
| [2] | Arbieto, A.; Matheus, C., A pasting lemma and some applications for conservative systems, Ergod. Theory Dyn. Syst., 27, 5, 1399-1417 (2007) · Zbl 1142.37025 · doi:10.1017/S014338570700017X |
| [3] | Arnaud, M-C, The generic symplectic C^1-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point, Ergod. Theory Dyn. Syst., 22, 6, 1621-1639 (2002) · Zbl 1030.37037 · doi:10.1017/S0143385702000706 |
| [4] | Arnaud, M.-C.: Le “closing lemma” en topologie C^1. Mm. Soc. Mat. Fr. (N.S.) 74, vi+120 (1998) · Zbl 0920.58039 |
| [5] | Arnold, V.I.: “Mathematical Methods of Classical Mechanics”. Springer (1978) · Zbl 0386.70001 |
| [6] | Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacy, P., On devaney’s definition of chaos, Am. Math. Montly, 99, 332-334 (1992) · Zbl 0758.58019 · doi:10.2307/2324899 |
| [7] | Bessa, M.; Duarte, P., Abundance of elliptic dynamics on conservative three-flows, Dyn. Syst. Intl. J., 23, 4, 409-424 (2008) · Zbl 1158.37010 · doi:10.1080/14689360802162872 |
| [8] | Bessa, M., Rocha, J.: Anosov versus Heterodimensional cycles: A C^1 dichotomy for conservative maps.http://cmup.fc.up.pt/cmup/bessa/ Preprint (2009) |
| [9] | Bochi, J., Genericity of zero Lyapunov exponents, Ergod. Theory Dyn. Syst., 22, 6, 1667-1696 (2002) · Zbl 1023.37006 · doi:10.1017/S0143385702001165 |
| [10] | Bochi, J.; Viana, M., The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. Math. 2, 161, 3, 1423-1485 (2005) · Zbl 1101.37039 |
| [11] | Bonatti, C.; Crovisier, S., Récurrence et généricité, Invent. Math., 158, 1, 33-104 (2004) · Zbl 1071.37015 · doi:10.1007/s00222-004-0368-1 |
| [12] | Bonatti, C., Díaz, L.J., Viana, M.: “Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective”. Encycl. Math. Sci. 102. Math. Phys. 3. Springer (2005) · Zbl 1060.37020 |
| [13] | Dacorogna, B.; Moser, J., On a partial differential equation involving the Jacobian determinant, Ann. Inst. Henri Poincaré, 7, 1, 1-26 (1990) · Zbl 0707.35041 |
| [14] | Devaney, R.: “An Introduction to Chaotic Dynamical Systems”. Addison-Wesley (1989) · Zbl 0695.58002 |
| [15] | Franks, J.: Anosov diffeomorphisms. Global Analysis (Proc. Sympos. Pure Math., vol. XIV, Berkeley, California, 1968) 61-93. American Mathematical Society, Providence, R.I. 1070 · Zbl 0207.54304 |
| [16] | Kuratowski, K.: “Topology, vol. 1”. Academic (1966) · Zbl 0158.40901 |
| [17] | Liao, SD, On the stability conjecture, Chinese Ann. Math., 1, 9-30 (1980) · Zbl 0449.58013 |
| [18] | Mañé, R., Persistent manifolds are normally hyperbolic, Bull. Am. Math. Soc., 80, 90-91 (1974) · Zbl 0276.58009 · doi:10.1090/S0002-9904-1974-13366-5 |
| [19] | Mañé, R., An ergodic closing lemma, Ann. Math., 116, 503-540 (1982) · Zbl 0511.58029 · doi:10.2307/2007021 |
| [20] | Mañé, R.: Oseledec’s theorem from the generic viewpoint. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pp. 1269-1276. Warsaw, 1984. PWN · Zbl 0584.58007 |
| [21] | Mañé, R.: The Lyapunov exponents of generic area preserving diffeomorphisms. In: International Conference on Dynamical Systems (Montevideo, 1995), vol. 362 Pitman Res. Notes Math. Ser., pp. 110-119. Longman, Harlow, 1996 · Zbl 0870.58083 |
| [22] | Mañé, R., Ergodic Theory and Differentiable Dynamics (1987), Berlin: Springer, Berlin · Zbl 0616.28007 |
| [23] | Mora, L.; Romero, N., Persistence of homoclinic tangencies for area-preserving maps, Ann. Fac. Sci. Toulouse Math., 6, 4, 711-725 (1997) · Zbl 0932.37033 |
| [24] | Newhouse, S., Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math., 99, 1061-1087 (1977) · Zbl 0379.58011 · doi:10.2307/2374000 |
| [25] | Oseledets, VI, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19, 197-231 (1968) · Zbl 0236.93034 |
| [26] | Palis, J., Takens, F.: “Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors”. Cambridge Studies in Advanced Mathematics, vol. 35. Cambridge University Press, Cambridge (1993) · Zbl 0790.58014 |
| [27] | Palis, J., Smale, S.: Structural stability theorems. 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, California, 1968) pp. 223-231. American Mathematical Society, Providence, R.I., 1968 · Zbl 0214.50702 |
| [28] | Palis, J., Open questions leading to a global perspective in dynamics, Nonlinearity, 21, 37-43 (2008) · Zbl 1147.37010 · doi:10.1088/0951-7715/21/4/T01 |
| [29] | Pliss, VA, On a conjecture of Smale, Differ. Uravnenija, 8, 268-282 (1972) · Zbl 0243.34077 |
| [30] | Pollicott, M.: “Lectures on Ergodic Theory and Pesin Theory on compact manifolds”. London Mathematical Society Lecture Notes Series, vol. 180. London Mathematical Society, Cambridge (1993) · Zbl 0772.58001 |
| [31] | Pugh, C.; Robinson, C., The C^1 closing lemma, including Hamiltonians, Ergod. Theory Dyn. Syst., 3, 261-313 (1983) · Zbl 0548.58012 · doi:10.1017/S0143385700001978 |
| [32] | Robinson, C., Generic properties of conservative systems, Am. J. Math., 92, 562-603 (1970) · Zbl 0212.56502 · doi:10.2307/2373361 |
| [33] | Robinson, C.: Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, 2nd edn. Stud. Adv. Math. CRC, Boca Raton, FL (1999) · Zbl 0914.58021 |
| [34] | Saghin, R.; Xia, Z., Partial Hyperbolicity or dense elliptical periodic points for C^1-generic symplectic diffeomorphisms, Trans. AMS, 358, 5119-5138 (2006) · Zbl 1210.37014 · doi:10.1090/S0002-9947-06-04171-7 |
| [35] | Shub, M., Global Stability of Dynamical Systems (1987), New York: Springer, New York · Zbl 0606.58003 |
| [36] | Takens, F., Homoclinic points in conservative systems, Invent. Math., 18, 267-292 (1972) · Zbl 0247.58007 · doi:10.1007/BF01389816 |
| [37] | Yoccoz, J. C., Travaux de Herman sur les Tores invariants, Astérisque, 206, 4, 311-344 (1992) · Zbl 0791.58044 |
| [38] | Zehnder, E.: Note on smoothing symplectic and volume-preserving diffeomorphisms. In: Proceedings III Latin American School of Mathematics, Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976, vol. 597 Lecture Notes in Math., pp. 828-854. Springer, Berlin (1977) · Zbl 0363.58004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
