How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency

Advances in Mathematics

Volumn 208, Issue 2, 2007, Pages 710-797

  Gorodetski, A ^a,b   Kaloshin, V ^a,c,d  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b INDEPENDENT UNIVERSITY OF MOSCOW   (Russian Federation)

^c INSTITUTE FOR ADVANCED STUDY   (United States)

^d AMERICAN INSTITUTE OF MATHEMATICS   (United States)

Author keywords

Homoclinic tangency; Infinitely many coexisting sinks; Newhouse domain; Non uniform hyperbolicity; Prevalence

Indexed keywords

EID: 33750625209     PISSN: 00018708     EISSN: 10902082     Source Type: Journal    
DOI: 10.1016/j.aim.2006.03.012     Document Type: Article

References (44)

1. Araujo V. Attractors and time averages for random maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 3 (2000) 307-369
2. Benedicks M., and Carleson L. The dynamics of the Hénon map. Ann. of Math. (2) 133 1 (1991) 73-169
3. Berezin I., and Zhidkov N. Computing Methods, vol. 1 (1965), Pergamon, Oxford, UK
4. Buzzard G. Infinitely many periodic attractors for holomorphic maps of 2 variables. Ann. of Math. (2) 145 2 (1997) 389-417
5. Christensen J. On sets of Haar measure zero in abelian Polish groups. Israel J. Math. 13 (1972) 255-260
6. Colli E. Infinitely many coexisting strange attractors. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 5 (1998) 539-579
7. Forness J., and Gavosto E. Existence of generic homoclinic tangencies for Henon mappings. J. Geom. Anal. 2 5 (1992) 429-444
8. Gavrilov N.K., and Shilnikov L.P. On the three-dimensional system close to a system with a structurally unstable homoclinic curve I, II. Math. USSR Sb. 17 (1972) 467-485
9. Gavrilov N.K., and Shilnikov L.P. On the three-dimensional system close to a system with a structurally unstable homoclinic curve I, II. Math. USSR Sb. 19 (1973) 139-156
10. Golubitsky M., and Guillemin V. Stable Mappings and Their Singularities. Grad. Texts in Math. vol. 14 (1973), Springer
11. Gonchenko S., and Shil'nikov L. On dynamical systems with structurally unstable homoclinic curves. Soviet Math. Dokl. 33 1 (1986) 234-238
12. Gonchenko S., Shil'nikov L., and Tuvaev D. On models with non-rough Poincare homoclinic curves. Phys. D 62 1-4 (1993) 1-14
13. Gonchenko S., Sten'kin O., and Tuvaev D. Complexity of homoclinic bifurcations and Ω-moduli. Internat. J. Bifur. Chaos 6 6 (1996) 969-989
14. Gonchenko S., Shil'nikov L., and Tuvaev D. Homoclinic tangencies of arbitrary order in Newhouse domains. Dynamical Systems. Moscow, 1998. Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz. vol. 67 (1999), Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow 69-128 (in Russian)
15. A. Gorodetski, B. Hunt, V. Kaloshin, Newton interpolation polynomials, discretization method, and certain prevalent properties in dynamical systems, Notes for the 2006 ICM talk, 29 pp. Available at www.its.caltech.edu/~kaloshin
16. Grigoriev A., and Yakovenko S. Topology of generic multijet preimages and blow-up via Newton interpolation. J. Differential Equations 150 (1998) 349-362
17. Henon M. A two-dimensional mapping with a strange attractor. Comm. Math. Phys. 50 1 (1976) 69-77
18. Hunt B., Sauer T., and Yorke J. Prevalence: A translation-invariant almost every for infinite-dimensional spaces. Bull. Amer. Math. Soc. 27 (1992) 217-238 Prevalence: An addendum
19. Hunt B., Sauer T., and Yorke J. Prevalence: A translation-invariant almost every for infinite-dimensional spaces. Bull. Amer. Math. Soc. 28 (1993) 306-307
20. Ilyashenko Yu., and Yakovenko S. Finitely smooth normal forms of local families of diffeomorphisms and vector fields. Russ. Math. Surv. 46 (1991) 1-43
21. Kaloshin V. Some prevalent properties of smooth dynamical systems. Proc. Steklov Math. Inst. 213 (1997) 123-151
22. V. Kaloshin, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II, preprint, 85 pp
23. Kaloshin V., and Hunt B. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I & II. Electron. Res. Announc. Amer. Math. Soc. 7 (2001) 17-27 & 28-36 (electronic)
24. V. Kaloshin, B. Hunt, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I, Ann. of Math., in press, 92 pp
25. Tedeschini-Lalli L., and Yorke J. How often do simple dynamical processes have infinitely many coexisting sinks?. Comm. Math. Phys. 106 4 (1986) 635-657
26. Mora L., and Viana M. Abundance of strange attractors. Acta Math. 171 1 (1993) 1-71
27. 2. Proc. Amer. Math. Soc. Sympos. Pure Math. 14 (1970) 191-202
28. Newhouse S. Diffeomorphisms with infinitely many sinks. Topology 13 (1974) 9-18
29. Newhouse S. The abundance of wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 50 (1979) 101-151
30. Newhouse S., and Palis J. Cycles and bifurcation theory. Astérisque 31 (1976) 44-140
31. Nusse H.E., and Tedeschini-Lalli L. Wild hyperbolic sets, yet no chance for the coexistence of infinitely many KLUS-simple Newhouse attracting sets. Comm. Math. Phys. 144 (1992) 429-442
32. Ott E., and Yorke J. Prevalence. Bull. Amer. Math. Soc. (N.S.) 42 3 (2005) 263-290
33. Palis J. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Géométrie complexe et systémes dynamiques. Orsay, 1995. Astérisque 261 (2000) xiii-xiv 335-347
34. Palis J., and Takens F. Hyperbolicity and the creation of homoclinic orbits. Ann. of Math. (2) 125 2 (1987) 337-374
35. Palis J., and Takens F. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (1993), Cambridge Univ. Press
36. Palis J., and Yoccoz J.-Ch. Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension. Acta Math. 172 1 (1994) 91-136
37. Palis J., and Yoccoz J.-Ch. Fers à cheval non uniformément hyperboliques engendrés par une bifurcation homocline et densité nulle des attracteurs (Non-uniformly hyperbolic horseshoes generated by homoclinic bifurcations and zero density of attractors). C. R. Acad. Sci. Paris Sér. I Math. 333 9 (2001) 867-871 (in French)
38. Pujals E., and Sambarino M. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2) 151 3 (2000) 961-1023
39. Robinson C. Bifurcations to infinitely many sinks. Comm. Math. Phys. 90 3 (1986) 433-459
40. Rios I.L. Unfolding homoclinic tangencies inside horseshoes: Hyperbolicity, fractal dimensions and persistent tangencies. Nonlinearity 14 3 (2001) 431-462
41. Thom R. Stabilite structurelle et morphogenese. Essai d'une theorie generale des modules. Math. Phys. Monogr. Ser. (1972), W.A. Benjamin, Reading, MA 362 pp. (in French)
42. Van Strien S. On the bifurcations creating horseshoes. In: Rand D., and Young L.-S. (Eds). Lecture Notes in Math. vol. 898 (1981), Springer, Berlin, Heidelberg, New York 316-351
43. Wang Q., and Young L.-S. Strange attractors with one direction of instability. Comm. Math. Phys. 218 1 (2001) 1-97
44. Yomdin Y. A quantitative version of the Kupka-Smale theorem. Ergodic Theory Dynam. Systems 5 (1985) 449-472