Structural stability of the inverse limit of endomorphisms
Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 5, pp. 1227-1253.

We prove that every endomorphism which satisfies Axiom A and the strong transversality conditions is C1-inverse limit structurally stable. These conditions were conjectured to be necessary and sufficient. This result is applied to the study of unfolding of some homoclinic tangencies. This also achieves a characterization of C1-inverse limit structurally stable covering maps.

Nous montrons qu'un endomorphisme a son extension naturelle qui est C1-structurellement stable s'il vérifie l'axiome A et la condition de transversalité forte. Ces conditions étaient conjecturées nécessaires et suffisantes. Ce résultat est appliqué à l'étude des déploiements des tangences homoclines. Aussi, cela accomplit la description des recouvrements dont l'extension naturelle est C1-structurellement stable.

DOI: 10.1016/j.anihpc.2016.10.001
Keywords: Inverse limit, Natural extension, Structural stability, Axiom A, Endomorphism, Strong transversality condition
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Berger, Pierre; Kocsard, Alejandro. Structural stability of the inverse limit of endomorphisms. Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 5, pp. 1227-1253. doi : 10.1016/j.anihpc.2016.10.001. http://www.numdam.org/articles/10.1016/j.anihpc.2016.10.001/

[1] Aoki, N.; Moriyasu, K.; Sumi, N. C1-maps having hyperbolic periodic points, Fundam. Math., Volume 169 (2001) no. 1, pp. 1–49 | DOI | MR | Zbl

[2] Berger, P.; Rovella, A. On the inverse limit stability of endomorphisms, Ann. non Linéaire IHP, Volume 30 (2013), pp. 463–475 | Numdam | MR | Zbl

[3] Graczyk, Jacek; Świątek, Grzegorz The real Fatou conjecture, Ann. Math. Stud., vol. 144, Princeton University Press, Princeton, NJ, 1998 | MR | Zbl

[4] Joly, R.; Raugel, G., Ann. Inst. Henri Poincaré, Volume vol. 27, Elsevier (2010), pp. 1397–1440 | Numdam | MR | Zbl

[5] Mikhail, Lyubich Dynamics of quadratic polynomials, I, Acta Math., Volume 178 (1997) no. 2, pp. 185–247 | MR | Zbl

[5] Mikhail, Lyubich Dynamics of quadratic polynomials, II, Acta Math., Volume 178 (1997) no. 2, pp. 247–297 | MR | Zbl

[6] Mañé, R. A proof of the C1 stability conjecture, Inst. Hautes Études Sci. Publ. Math. (1988) no. 66, pp. 161–210 | Numdam | MR | Zbl

[7] Mather, J. Notes on topological stability, Bull. Am. Math. Soc. (N.S.), Volume 49 (2012) no. 4, pp. 475–506 | DOI | MR | Zbl

[8] Mora, Leonardo Homoclinic bifurcations, fat attractors and invariant curves, Discrete Contin. Dyn. Syst., Volume 9 (2003) no. 5, pp. 1133–1148 | MR | Zbl

[9] Mañé, R.; Pugh, C. Dynamical Systems, Lect. Notes Math., Volume vol. 468, Springer, Berlin (1975), pp. 175–184 (Warwick, 1974) | DOI | MR | Zbl

[10] Palis, Jacob On Morse–Smale dynamical systems, Topology, Volume 8 (1969) no. 4, pp. 385–404 | MR | Zbl

[11] Przytycki, Feliks Anosov endomorphisms, Stud. Math., Volume 58 (1976) no. 3, pp. 249–285 | MR | Zbl

[12] Przytycki, F. On Ω-stability and structural stability of endomorphisms satisfying Axiom A, Stud. Math., Volume 60 (1977) no. 1, pp. 61–77 | DOI | MR | Zbl

[13] Palis, J.; Smale, S. Global Analysis, Proc. Symp. Pure Math., Volume vol. XIV, Amer. Math. Soc., Providence, RI (1970), pp. 223–231 (Berkeley, CA, 1968) | DOI | MR

[14] Palis, Jacob; Takens, Floris Fractal dimensions and infinitely many attractors, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Camb. Stud. Adv. Math., vol. 35, Cambridge University Press, Cambridge, 1993 | MR | Zbl

[15] Quandt, J. Stability of Anosov maps, Proc. Am. Math. Soc., Volume 104 (1988) no. 1, pp. 303–309 | DOI | MR | Zbl

[16] Quandt, J. On inverse limit stability for maps, J. Differ. Equ., Volume 79 (1989) no. 2, pp. 316–339 | DOI | MR | Zbl

[17] Robbin, J.W. A structural stability theorem, Ann. Math. (2), Volume 94 (1971), pp. 447–493 | DOI | MR | Zbl

[18] Robinson, C. Structural stability of C1 diffeomorphisms, J. Differ. Equ., Volume 22 (1976) no. 1, pp. 28–73 | DOI | MR | Zbl

[19] Smale, S. Differentiable dynamical systems, Bull. Am. Math. Soc., Volume 73 (1967), pp. 747–817 | DOI | MR | Zbl

[20] Sullivan, Dennis Topological Methods in Modern Mathematics, Publish or Perish, Houston, TX (1991), pp. 543–564 (Stony Brook, NY, 1991) | MR | Zbl

[21] Tatjer, Joan Carles Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergod. Theory Dyn. Syst., Volume 21 (2001) no. 1, pp. 249–302 | MR | Zbl

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