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# Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms, I.

Kaloshin, Vadim Yu. and Hunt, Brian R. (2007) Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms, I. Annals of Mathematics, 165 (1). pp. 89-170. ISSN 0003-486X. https://resolver.caltech.edu/CaltechAUTHORS:KALaom07

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## Abstract

For diffeomorphisms of smooth compact finite-dimensional 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 $\rho, \delta > 0$, there is a prevalent set of $C^{1+\rho}$ (or smoother) diffeomorphisms for which the number of periodic $n$ points is bounded above by $\exp(C n^{1+\delta})$ for some $C$ independent of $n$. We also obtain a related bound on the decay of hyperbolicity of the periodic points as a function of $n$, and obtain the same results for $1$-dimensional endomorphisms. 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 show this to be a subtle and complex phenomenon, reminiscent of KAM theory. Here in Part I we state our results and describe the methods we use. We complete most of the proof in the $1$-dimensional $C^2$-smooth case and outline the remaining steps, deferred to Part II, that are needed to establish the general case. The novel feature of the approach we develop in this paper is the introduction of Newton Interpolation Polynomials as a tool for perturbing trajectories of iterated maps.

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Additional Information:2007 © Princeton University (Mathematics Department) Acknowledgments. It is great pleasure to thank the thesis advisor of the first author, John Mather, who regularly spent hours listening to oral expositions of various parts of the proof for nearly two years [4]. Without his patience and support this project would never have been completed. The authors are truly grateful to Anton Gorodetski, Giovanni Forni, and an anonymous referee who read the manuscript carefully and have made many useful remarks. The first author is grateful to Anatole Katok for providing an opportunity to give a minicourse on the subject of this paper at Penn State University during the fall of 2000. The authors have profited from conversations with Carlo Carminati, Bill Cowieson, Dima Dolgopyat, Anatole Katok, Michael Lyubich, Michael Shub, Yakov Sinai, Marcelo Viana, Jean-Christophe Yoccoz, Lai-Sang Young, and many others. The first author thanks the Institute for Physical Science and Technology, University of Maryland and, in particular, James Yorke for their hospitality. The second author is grateful in turn to the Institute for Advanced Study at Princeton for its hospitality. The first author acknowledges the support of a Sloan dissertation fellowship during his final year at Princeton, when significant parts of the work were done. The first author is supported by NSF-grant DMS-0300229 and the second author by NSF grant DMS-0104087.
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Deposited On:15 Aug 2007