[{"publication":"Electronic Research Announcements of the American Mathematical Society","page":"28-36","publisher":"American Mathematical Society","fulldoi":"https://doi.org/10.1090/s1079-6762-01-00091-9","keyword":["General Mathematics"],"day":"24","year":"2001","doi":"10.1090/s1079-6762-01-00091-9","volume":7,"abstract":[{"text":"We continue the previous article's discussion of bounds, for prevalent diffeomorphisms of smooth compact manifolds, on the growth of the number of periodic points and the decay of their hyperbolicity as a function of their period $n$. In that article we reduced the main results to a problem, for certain families of diffeomorphisms, of bounding the measure of parameter values for which the diffeomorphism has (for a given period $n$) an almost periodic point that is almost nonhyperbolic. We also formulated our results for $1$-dimensional endomorphisms on a compact interval. In this article we describe some of the main techniques involved and outline the rest of the proof. To simplify notation, we concentrate primarily on the $1$-dimensional case.","lang":"eng"}],"title":"A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II","intvolume":"         7","date_published":"2001-04-24T00:00:00Z","article_type":"original","publication_identifier":{"issn":["1079-6762"]},"_id":"8521","date_created":"2020-09-18T10:49:43Z","date_updated":"2021-01-12T08:19:51Z","status":"public","language":[{"iso":"eng"}],"issue":"5","publication_status":"published","author":[{"orcid":"0000-0002-6051-2628","last_name":"Kaloshin","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","full_name":"Kaloshin, Vadim","first_name":"Vadim"},{"first_name":"Brian R.","full_name":"Hunt, Brian R.","last_name":"Hunt"}],"type":"journal_article","extern":"1","quality_controlled":"1","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"chicago":"Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms II.” <i>Electronic Research Announcements of the American Mathematical Society</i>. American Mathematical Society, 2001. <a href=\"https://doi.org/10.1090/s1079-6762-01-00091-9\">https://doi.org/10.1090/s1079-6762-01-00091-9</a>.","ieee":"V. Kaloshin and B. R. Hunt, “A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II,” <i>Electronic Research Announcements of the American Mathematical Society</i>, vol. 7, no. 5. American Mathematical Society, pp. 28–36, 2001.","apa":"Kaloshin, V., &#38; Hunt, B. R. (2001). A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II. <i>Electronic Research Announcements of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/s1079-6762-01-00091-9\">https://doi.org/10.1090/s1079-6762-01-00091-9</a>","ama":"Kaloshin V, Hunt BR. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II. <i>Electronic Research Announcements of the American Mathematical Society</i>. 2001;7(5):28-36. doi:<a href=\"https://doi.org/10.1090/s1079-6762-01-00091-9\">10.1090/s1079-6762-01-00091-9</a>","ista":"Kaloshin V, Hunt BR. 2001. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II. Electronic Research Announcements of the American Mathematical Society. 7(5), 28–36.","short":"V. Kaloshin, B.R. Hunt, Electronic Research Announcements of the American Mathematical Society 7 (2001) 28–36.","mla":"Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms II.” <i>Electronic Research Announcements of the American Mathematical Society</i>, vol. 7, no. 5, American Mathematical Society, 2001, pp. 28–36, doi:<a href=\"https://doi.org/10.1090/s1079-6762-01-00091-9\">10.1090/s1079-6762-01-00091-9</a>."},"oa_version":"None","month":"04"},{"date_created":"2020-09-18T10:49:56Z","date_updated":"2021-01-12T08:19:51Z","issue":"4","language":[{"iso":"eng"}],"status":"public","quality_controlled":"1","extern":"1","author":[{"first_name":"Vadim","full_name":"Kaloshin, Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","last_name":"Kaloshin","orcid":"0000-0002-6051-2628"},{"first_name":"Brian R.","full_name":"Hunt, Brian R.","last_name":"Hunt"}],"type":"journal_article","publication_status":"published","month":"04","oa_version":"None","citation":{"mla":"Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms I.” <i>Electronic Research Announcements of the American Mathematical Society</i>, vol. 7, no. 4, American Mathematical Society, 2001, pp. 17–27, doi:<a href=\"https://doi.org/10.1090/s1079-6762-01-00090-7\">10.1090/s1079-6762-01-00090-7</a>.","ista":"Kaloshin V, Hunt BR. 2001. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I. Electronic Research Announcements of the American Mathematical Society. 7(4), 17–27.","short":"V. Kaloshin, B.R. Hunt, Electronic Research Announcements of the American Mathematical Society 7 (2001) 17–27.","apa":"Kaloshin, V., &#38; Hunt, B. R. (2001). A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I. <i>Electronic Research Announcements of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/s1079-6762-01-00090-7\">https://doi.org/10.1090/s1079-6762-01-00090-7</a>","ieee":"V. Kaloshin and B. R. Hunt, “A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I,” <i>Electronic Research Announcements of the American Mathematical Society</i>, vol. 7, no. 4. American Mathematical Society, pp. 17–27, 2001.","ama":"Kaloshin V, Hunt BR. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I. <i>Electronic Research Announcements of the American Mathematical Society</i>. 2001;7(4):17-27. doi:<a href=\"https://doi.org/10.1090/s1079-6762-01-00090-7\">10.1090/s1079-6762-01-00090-7</a>","chicago":"Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms I.” <i>Electronic Research Announcements of the American Mathematical Society</i>. American Mathematical Society, 2001. <a href=\"https://doi.org/10.1090/s1079-6762-01-00090-7\">https://doi.org/10.1090/s1079-6762-01-00090-7</a>."},"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication":"Electronic Research Announcements of the American Mathematical Society","page":"17-27","publisher":"American Mathematical Society","abstract":[{"lang":"eng","text":"For diffeomorphisms of smooth compact manifolds, we consider the problem of how fast the number of periodic points with period $n$grows as a function of $n$. In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for $C^2$ or smoother diffeomorphisms. In the present work we show that, by contrast, for a measure-theoretic notion of genericity we call ``prevalence'', the growth is not much faster than exponential. Specifically, we show that for each $\\delta > 0$, there is a prevalent set of ( $C^{1+\\rho}$ or smoother) diffeomorphisms for which the number of period $n$ points is bounded above by $\\operatorname{exp}(C n^{1+\\delta})$ for some $C$ independent of $n$. We also obtain a related bound on the decay of the hyperbolicity of the periodic points as a function of $n$. The contrast between topologically generic and measure-theoretically generic behavior for the growth of the number of periodic points and the decay of their hyperbolicity shows this to be a subtle and complex phenomenon, reminiscent of KAM theory."}],"volume":7,"year":"2001","doi":"10.1090/s1079-6762-01-00090-7","fulldoi":"https://doi.org/10.1090/s1079-6762-01-00090-7","keyword":["General Mathematics"],"day":"18","date_published":"2001-04-18T00:00:00Z","article_type":"original","title":"A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I","intvolume":"         7","_id":"8522","publication_identifier":{"issn":["1079-6762"]}}]