Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale  [ Stable homoclinic tangencies for hyperbolic sets of large fractal dimension ]
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 1, p. 1-68

Let F 0 be a surface diffeomorphism with two horseshoes Λ,Λ ' such that W s Λ and W u Λ ' have a quadratic tangency at a point q. We show that, if the sum of the transverse dimension of W s Λ and W u Λ ' is larger than one, the set of diffeomorphisms close to F 0 such that W s Λ and W u Λ ' have a stable tangency near q has positive density at F 0 .

Soit F 0 un difféomorphisme d’une surface possédant deux fers à cheval Λ,Λ ' tels que W s Λ et W u Λ ' aient en un point q une tangence quadratique isolée. Nous montrons que, si la somme des dimensions transverses de W s Λ et W u Λ ' est strictement plus grande que 1, les difféomorphismes voisins de F 0 tels que W s Λ et W u Λ ' soient stablement tangents au voisinage de q forment une partie de densité inférieure strictement positive en F 0 .

DOI : https://doi.org/10.24033/asens.2115
Classification:  37D05,  37D20,  37E30
Keywords: homoclinic bifurcation, homoclinic tangency, horseshoe, fractal dimension
@article{ASENS_2010_4_43_1_1_0,
     author = {Moreira, Carlos Gustavo and Yoccoz, Jean-Christophe},
     title = {Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {4e s{\'e}rie, 43},
     number = {1},
     year = {2010},
     pages = {1-68},
     doi = {10.24033/asens.2115},
     zbl = {1200.37020},
     mrnumber = {2583264},
     language = {fr},
     url = {http://www.numdam.org/item/ASENS_2010_4_43_1_1_0}
}
Moreira, Carlos Gustavo; Yoccoz, Jean-Christophe. Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 1, pp. 1-68. doi : 10.24033/asens.2115. http://www.numdam.org/item/ASENS_2010_4_43_1_1_0/

[1] M. J. Hall, On the sum and product of continued fractions, Ann. of Math. 48 (1947), 966-993. | MR 22568 | Zbl 0030.02201

[2] M. W. Hirsch, C. C. Pugh & M. Shub, Invariant manifolds, Lecture Notes in Math. 583, Springer, 1977. | MR 501173 | Zbl 0355.58009

[3] R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153-155. | MR 248779 | Zbl 0165.37404

[4] J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. 4 (1954), 257-302. | MR 63439 | Zbl 0056.05504

[5] C. G. Moreira, Stable intersections of Cantor sets and homoclinic bifurcations, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 741-781. | Numdam | MR 1420497 | Zbl 0865.58035

[6] C. G. Moreira & J.-C. Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions, Ann. of Math. 154 (2001), 45-96. | MR 1847588 | Zbl 1195.37015

[7] S. E. Newhouse, Nondensity of axiom A(a) on S 2 , in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., 1970, 191-202. | MR 277005 | Zbl 0206.25801

[8] J. Palis & F. Takens, Hyperbolicity and the creation of homoclinic orbits, Ann. of Math. 125 (1987), 337-374. | MR 881272 | Zbl 0641.58029

[9] J. Palis & F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Math. 35, Cambridge Univ. Press, 1993. | MR 1237641 | Zbl 0790.58014

[10] J. Palis & J.-C. Yoccoz, Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension, Acta Math. 172 (1994), 91-136. | MR 1263999 | Zbl 0801.58035

[11] E. R. Pujals & M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. 151 (2000), 961-1023. | MR 1779562 | Zbl 0959.37040

[12] D. Sullivan, Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets, in The mathematical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math. 48, Amer. Math. Soc., 1988, 15-23. | MR 974329 | Zbl 0665.58027