TAILIEUCHUNG - Đề tài "Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms I "

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 (., 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. | Annals of Mathematics Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms I By Vadim Yu. Kaloshin and Brian R. Hunt Annals of Mathematics 165 2007 89 170 Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms I By Vadim Yu. Kaloshin and Brian R. Hunt 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 . 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 C2 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 p ỗ 0 there is a prevalent set of C1 p or smoother diffeomorphisms for which the number of periodic n points is bounded above by exp Cn1 ổ 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 C2-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. Table of contents 1. A problem of the growth of

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