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

• 成果类型: Article
• 署名作者: Kaloshin, Vadim Yu.; Hunt, Brian R.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2007.165.89
• 发表日期: 2007
• 页码: 89-170
• 关键词: superexponential growth; invariant manifolds; THEOREM; Orbits; FLOWS

摘要

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 p, delta > 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(Cn(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 I-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.