Persistence of stratifications of normally expanded laminations
Mémoires de la Société Mathématique de France, no. 134 (2013) , 113 p.

Ce travail s’inscrit dans le prolongement de celui de Hirsch-Pugh-Shub (HPS) sur la persistance des laminations normalement hyperboliques, et implique plusieurs théorèmes de stabilité structurelle. On généralise le concepte de lamination par une nouvelle catégorie d’objets : les stratifications de laminations. Il s’agit de stratifications, dont les strates sont des laminations. On propose alors un théorème assurant la persistance de certaines stratifications dont chaque strate est une lamination normalement dilatée. La dynamique est un C r -endomorphisme d’une variété (qui n’est donc pas forcément inversible et qui peut avoir des points critiques). La persistance signifie que toute C r -perturbation de la dynamique préserve une stratification C r -proche. Quand la stratification est formée d’une unique strate, le théoreme principal donne la persistance des laminations normalement dilatées par un endomorphisme, et implique ainsi le théorème de HPS. Une autre application de ce théorème est la persistance des variétés à bord ou à coins normalement dilatés. Beaucoup examples sont donnés facilement en dynamique produit. Aussi les difféomorphismes vérifiant l’axiome A et la condition de transversalité forte (ATF) possèdent deux stratifications de laminations canoniques : celle dont les strates sont les ensembles stables (resp. instables) de ses pièces basiques. Ainsi, notre théorème implique la persistance de certaines laminations “normalement ATF” qui ne sont pas normalement hyperboliques et d’autres théorèmes de stabilité structurelle.

This manuscript complements the Hirsch-Pugh-Shub (HPS) theory on persistence of normally hyperbolic laminations and implies several structural stability theorems. We generalize the concept of lamination by defining a new object: the stratification of laminations. It is a stratification whose strata are laminations. The main theorem implies the persistence of some stratifications whose strata are normally expanded. The dynamics is a C r -endomorphism of a manifold (which is possibly not invertible and with critical points). The persistence means that any C r -perturbation of the dynamics preserves a C r -close stratification. If the stratification consists of a single stratum, the main theorem implies the persistence of normally expanded laminations by endomorphisms, and hence implies HPS theorem. Another application of this theorem is the persistence, as stratifications, of submanifolds with boundary or corners normally expanded. Several examples are also given in product dynamics. As diffeomorphisms that satisfy axiom A and the strong transversality condition (AS) defines canonically two stratifications of laminations: the stratification whose strata are the (un)stable sets of basic pieces of the spectral decomposition. The main theorem implies the persistence of some “normally AS” laminations which are not normally hyperbolic and other structural stability theorems.

DOI : 10.24033/msmf.444
Mots clés : Laminations, Stratifications, Structural Stability, Persistence, Hyperbolic Dynamics, Endomorphisms, Axiom A, Product dynamics.
@book{MSMF_2013_2_134__1_0,
     author = {Berger, Pierre},
     title = {Persistence of stratifications of normally expanded laminations},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {134},
     year = {2013},
     doi = {10.24033/msmf.444},
     mrnumber = {3154396},
     zbl = {06318198},
     language = {en},
     url = {http://archive.numdam.org/item/MSMF_2013_2_134__1_0/}
}
TY  - BOOK
AU  - Berger, Pierre
TI  - Persistence of stratifications of normally expanded laminations
T3  - Mémoires de la Société Mathématique de France
PY  - 2013
IS  - 134
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/MSMF_2013_2_134__1_0/
DO  - 10.24033/msmf.444
LA  - en
ID  - MSMF_2013_2_134__1_0
ER  - 
%0 Book
%A Berger, Pierre
%T Persistence of stratifications of normally expanded laminations
%S Mémoires de la Société Mathématique de France
%D 2013
%N 134
%I Société mathématique de France
%U http://archive.numdam.org/item/MSMF_2013_2_134__1_0/
%R 10.24033/msmf.444
%G en
%F MSMF_2013_2_134__1_0
Berger, Pierre. Persistence of stratifications of normally expanded laminations. Mémoires de la Société Mathématique de France, Série 2, no. 134 (2013), 113 p. doi : 10.24033/msmf.444. http://numdam.org/item/MSMF_2013_2_134__1_0/

[1] V. I. ArnolʼD, S. M. Guseĭn-Zade & A. N. VarchenkoSingularities of differentiable maps. Vol. I, Monographs in Mathematics, vol. 82, Birkhäuser Boston Inc., Boston, MA, 1985, The classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous and Mark Reynolds.

[2] K. Bekka« C-régularité et trivialité topologique », in Singularity theory and its applications, Part I (Coventry, 1988/1989), Lecture Notes in Math., vol. 1462, Springer, Berlin, 1991, p. 42–62. | MR | Zbl

[3] P. Berger« Persistence des stratifications de laminations normalement dilatées », PhD Thesis, Université Paris XI (2007).

[4] —, « Persistent bundles over a two dimensional compact set », ArXiv e-prints (2009).

[5] —, « Persistence of laminations », Bull Braz Math Soc, New series 41(2) (2010), p. 259–319. | MR | Zbl

[6] —, « Persistance des sous variétés à bord et à coins », Ann. Inst. Fourier 61(1) (2011), p. 79–104. | MR | Numdam

[7] P. Berger & A. Rovella« On the inverse limit stability of endomorphisms », to appear in Ann. Inst. H. Poincaré (C) Non Linear Analysis; http://arxiv.org/abs/1006.4302. | MR | Zbl | Numdam

[8] C. Bonatti, L. J. Díaz & M. VianaDynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005, A global geometric and probabilistic perspective, Mathematical Physics, III. | MR | Zbl

[9] J. Buzzi, O. Sester & M. Tsujii« Weakly expanding skew-products of quadratic maps », Ergodic Theory Dynam. Systems 23 (2003), no. 5, p. 1401–1414. | MR | Zbl

[10] J. Cerf« Topologie de certains espaces de plongements », Bull. Soc. Math. France 89 (1961), p. 227–380. | MR | EuDML | Zbl | Numdam

[11] A. Douady« Variétés à bord anguleux et voisinages tubulaires », in Séminaire Henri Cartan, 1961/62, Exp. 1, Secrétariat mathématique, Paris, 1961/1962, p. 11. | MR | EuDML | Zbl | Numdam

[12] É. Ghys« Laminations par surfaces de Riemann », in Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, p. ix, xi, 49–95. | Zbl

[13] N. Gourmelon« Adapted metrics for dominated splittings », Ergodic Theory Dynam. Systems 27 (2007), no. 6, p. 1839–1849. | MR | Zbl

[14] J. Graczyk & G. ŚwiątekThe real Fatou conjecture, Annals of Mathematics Studies, vol. 144, Princeton University Press, Princeton, NJ, 1998. | MR | Zbl

[15] M. W. Hirsch, C. C. Pugh & M. ShubInvariant manifolds, Springer-Verlag, Berlin, 1977, Lecture Notes in Mathematics, Vol. 583. | MR | Zbl

[16] M. W. HirschDifferential topology, Springer-Verlag, New York, 1976, Graduate Texts in Mathematics, No. 33. | MR | Zbl

[17] H. Karcher« Riemannian center of mass and mollifier smoothing », Comm. Pure Appl. Math. 30 (1977), no. 5, p. 509–541. | MR | Zbl

[18] M. Lyubich« Dynamics of quadratic polynomials. I, II », Acta Math. 178 (1997), no. 2, p. 185–247, 247–297. | MR | Zbl

[19] R. Mañé« Persistent manifolds are normally hyperbolic », Trans. Amer. Math. Soc. 246 (1978), p. 261–283. | MR | Zbl

[20] —, « A proof of the C 1 stability conjecture », Inst. Hautes Études Sci. Publ. Math. (1988), no. 66, p. 161–210. | Zbl | Numdam

[21] J. N. Mather« Stratifications and mappings », in Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, p. 195–232. | MR

[22] W. De Melo« Structural stability of diffeomorphisms on two-manifolds », Invent. Math. 21 (1973), p. 233–246. | MR | EuDML | Zbl

[23] P. W. MichorManifolds of differentiable mappings, Shiva Mathematics Series, vol. 3, Shiva Publishing Ltd., Nantwich, 1980. | MR | Zbl

[24] J. MilnorDynamics in one complex variable, third éd., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. | MR

[25] C. Murolo & D. Trotman« Semidifférentiabilité et version lisse de la conjecture de fibration de whitney », Advanced Studies in Pure Mathematics (2006), no. 43, p. 271–309. | MR | Zbl

[26] J. Palis & S. Smale« Structural stability theorems », in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, p. 223–231. | MR | Zbl

[27] J. Robbin« A structural stability theorem », Ann. of Math. (2) 94 (1971), p. 447–493. | MR | Zbl

[28] C. Robinson« Structural stability of C 1 diffeomorphisms », J. Differential Equations 22 (1976), no. 1, p. 28–73. | MR | Zbl

[29] F. Rodriguez Hertz, M. A. Rodriguez Hertz & R. Ures« A survey of partially hyperbolic dynamics », in Partially hyperbolic dynamics, laminations, and Teichmüller flow, Fields Inst. Commun., vol. 51, Amer. Math. Soc., Providence, RI, 2007, p. 35–87. | MR | Zbl

[30] M. Shub« Endomorphisms of compact differentiable manifolds », Amer. J. Math. 91 (1969), p. 175–199. | MR | Zbl

[31] —, Stabilité globale des systèmes dynamiques, Astérisque, vol. 56, Société Mathématique de France, Paris, 1978, With an English preface and summary. | Zbl | Numdam

[32] S. Smale« Differentiable dynamical systems », Bull. Amer. Math. Soc. 73 (1967), p. 747–817. | MR | Zbl

[33] R. Thom« Local topological properties of differentiable mappings », in Differential Analysis, Bombay Colloq., Oxford Univ. Press, London, 1964, p. 191–202. | MR

[34] D. J. A. Trotman« Geometric versions of Whitney regularity for smooth stratifications », Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, p. 453–463. | MR | EuDML | Zbl | Numdam

[35] M. Viana« Multidimensional nonhyperbolic attractors », Inst. Hautes Études Sci. Publ. Math. (1997), no. 85, p. 63–96. | MR | EuDML | Zbl

[36] H. Whitney« Local properties of analytic varieties », in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965, p. 205–244. | MR | Zbl

[37] —, « Tangents to an analytic variety », Ann. of Math. (2) 81 (1965), p. 496–549. | MR | Zbl

[38] J.-C. Yoccoz« Introduction to hyperbolic dynamics », in Real and complex dynamical systems (Hillerød, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 464, Kluwer Acad. Publ., Dordrecht, 1995, p. 265–291. | MR | Zbl

Cité par Sources :