Stable homoclinic tangencies for hyperbolic sets of large fractal dimension

Carlos Gustavo Moreira; Jean-Christophe Yoccoz

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 1, page 1-68
  • ISSN: 0012-9593

Abstract

top
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 .

How to cite

top

Moreira, Carlos Gustavo, and Yoccoz, Jean-Christophe. "Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale." Annales scientifiques de l'École Normale Supérieure 43.1 (2010): 1-68. <http://eudml.org/doc/272127>.

@article{Moreira2010,
abstract = {Soit $F_0$ un difféomorphisme d’une surface possédant deux fers à cheval $\Lambda , \Lambda ^\{\prime \}$ tels que $W^s \Lambda $ et $W^u \Lambda ^\{\prime \}$ aient en un point $q$ une tangence quadratique isolée. Nous montrons que, si la somme des dimensions transverses de $W^s \Lambda $ et $W^u \Lambda ^\{\prime \}$ est strictement plus grande que 1, les difféomorphismes voisins de $F_0$ tels que $W^s \Lambda $ et $W^u \Lambda ^\{\prime \}$ soient stablement tangents au voisinage de $q$ forment une partie de densité inférieure strictement positive en $F_0$.},
author = {Moreira, Carlos Gustavo, Yoccoz, Jean-Christophe},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {homoclinic bifurcation; homoclinic tangency; horseshoe; fractal dimension},
language = {fre},
number = {1},
pages = {1-68},
publisher = {Société mathématique de France},
title = {Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale},
url = {http://eudml.org/doc/272127},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Moreira, Carlos Gustavo
AU - Yoccoz, Jean-Christophe
TI - Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 1
SP - 1
EP - 68
AB - Soit $F_0$ un difféomorphisme d’une surface possédant deux fers à cheval $\Lambda , \Lambda ^{\prime }$ tels que $W^s \Lambda $ et $W^u \Lambda ^{\prime }$ aient en un point $q$ une tangence quadratique isolée. Nous montrons que, si la somme des dimensions transverses de $W^s \Lambda $ et $W^u \Lambda ^{\prime }$ est strictement plus grande que 1, les difféomorphismes voisins de $F_0$ tels que $W^s \Lambda $ et $W^u \Lambda ^{\prime }$ soient stablement tangents au voisinage de $q$ forment une partie de densité inférieure strictement positive en $F_0$.
LA - fre
KW - homoclinic bifurcation; homoclinic tangency; horseshoe; fractal dimension
UR - http://eudml.org/doc/272127
ER -

References

top
  1. [1] M. J. Hall, On the sum and product of continued fractions, Ann. of Math.48 (1947), 966–993. Zbl0030.02201MR22568
  2. [2] M. W. Hirsch, C. C. Pugh & M. Shub, Invariant manifolds, Lecture Notes in Math. 583, Springer, 1977. Zbl0355.58009MR501173
  3. [3] R. Kaufman, On Hausdorff dimension of projections, Mathematika15 (1968), 153–155. Zbl0165.37404MR248779
  4. [4] J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc.4 (1954), 257–302. Zbl0056.05504MR63439
  5. [5] C. G. Moreira, Stable intersections of Cantor sets and homoclinic bifurcations, Ann. Inst. H. Poincaré Anal. Non Linéaire13 (1996), 741–781. Zbl0865.58035MR1420497
  6. [6] C. G. Moreira & J.-C. Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions, Ann. of Math.154 (2001), 45–96. Zbl1195.37015MR1847588
  7. [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. Zbl0206.25801MR277005
  8. [8] J. Palis & F. Takens, Hyperbolicity and the creation of homoclinic orbits, Ann. of Math.125 (1987), 337–374. Zbl0641.58029MR881272
  9. [9] J. Palis & F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Math. 35, Cambridge Univ. Press, 1993. Zbl0790.58014MR1237641
  10. [10] J. Palis & J.-C. Yoccoz, Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension, Acta Math.172 (1994), 91–136. Zbl0801.58035MR1263999
  11. [11] E. R. Pujals & M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math.151 (2000), 961–1023. Zbl0959.37040MR1779562
  12. [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. Zbl0665.58027MR974329

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.