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

Vadim Kaloshin ; Brian R. Hunt¹

¹ Department of Mathematics, University of Maryland, College Park, MD 20742, United States

Annals of mathematics, Tome 165 (2007) no. 1, pp. 89-170

Résumé

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.

Détail

DOI : 10.4007/annals.2007.165.89