{"article_type":"original","page":"710-797","publication_identifier":{"issn":["0001-8708"]},"issue":"2","year":"2007","type":"journal_article","day":"30","publication_status":"published","oa_version":"None","quality_controlled":"1","date_updated":"2021-01-12T08:19:47Z","publisher":"Elsevier","publication":"Advances in Mathematics","_id":"8511","status":"public","volume":208,"abstract":[{"text":"Here we study an amazing phenomenon discovered by Newhouse [S. Newhouse, Non-density of Axiom A(a) on S2, in: Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., 1970, pp. 191–202; S. Newhouse,\r\nDiffeomorphisms with infinitely many sinks, Topology 13 (1974) 9–18; S. Newhouse, The abundance of\r\nwild hyperbolic sets and nonsmooth stable sets of diffeomorphisms, Publ. Math. Inst. Hautes Études Sci.\r\n50 (1979) 101–151]. It turns out that in the space of Cr smooth diffeomorphisms Diffr(M) of a compact\r\nsurface M there is an open set U such that a Baire generic diffeomorphism f ∈ U has infinitely many coexisting sinks. In this paper we make a step towards understanding “how often does a surface diffeomorphism\r\nhave infinitely many sinks.” Our main result roughly says that with probability one for any positive D a\r\nsurface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by D or those\r\nwhose period is relatively large compared to its cyclicity. It verifies a particular case of Palis’ Conjecture\r\nsaying that even though diffeomorphisms with infinitely many coexisting sinks are Baire generic, they have\r\nprobability zero.\r\nOne of the key points of the proof is an application of Newton Interpolation Polynomials to study the dynamics initiated in [V. Kaloshin, B. Hunt, A stretched exponential bound on the rate of growth of the number\r\nof periodic points for prevalent diffeomorphisms I, Ann. of Math., in press, 92 pp.; V. Kaloshin, A stretched\r\nexponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II,\r\npreprint, 85 pp.].","lang":"eng"}],"date_published":"2007-01-30T00:00:00Z","citation":{"ama":"Gorodetski A, Kaloshin V. How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency. <i>Advances in Mathematics</i>. 2007;208(2):710-797. doi:<a href=\"https://doi.org/10.1016/j.aim.2006.03.012\">10.1016/j.aim.2006.03.012</a>","apa":"Gorodetski, A., &#38; Kaloshin, V. (2007). How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2006.03.012\">https://doi.org/10.1016/j.aim.2006.03.012</a>","chicago":"Gorodetski, A., and Vadim Kaloshin. “How Often Surface Diffeomorphisms Have Infinitely Many Sinks and Hyperbolicity of Periodic Points near a Homoclinic Tangency.” <i>Advances in Mathematics</i>. Elsevier, 2007. <a href=\"https://doi.org/10.1016/j.aim.2006.03.012\">https://doi.org/10.1016/j.aim.2006.03.012</a>.","ista":"Gorodetski A, Kaloshin V. 2007. How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency. Advances in Mathematics. 208(2), 710–797.","ieee":"A. Gorodetski and V. Kaloshin, “How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency,” <i>Advances in Mathematics</i>, vol. 208, no. 2. Elsevier, pp. 710–797, 2007.","short":"A. Gorodetski, V. Kaloshin, Advances in Mathematics 208 (2007) 710–797.","mla":"Gorodetski, A., and Vadim Kaloshin. “How Often Surface Diffeomorphisms Have Infinitely Many Sinks and Hyperbolicity of Periodic Points near a Homoclinic Tangency.” <i>Advances in Mathematics</i>, vol. 208, no. 2, Elsevier, 2007, pp. 710–97, doi:<a href=\"https://doi.org/10.1016/j.aim.2006.03.012\">10.1016/j.aim.2006.03.012</a>."},"date_created":"2020-09-18T10:48:27Z","keyword":["General Mathematics"],"doi":"10.1016/j.aim.2006.03.012","article_processing_charge":"No","month":"01","author":[{"full_name":"Gorodetski, A.","last_name":"Gorodetski","first_name":"A."},{"last_name":"Kaloshin","orcid":"0000-0002-6051-2628","full_name":"Kaloshin, Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","first_name":"Vadim"}],"extern":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"intvolume":"       208","title":"How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency"}