@article{8521,
  abstract     = {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.},
  author       = {Kaloshin, Vadim and Hunt, Brian R.},
  issn         = {1079-6762},
  journal      = {Electronic Research Announcements of the American Mathematical Society},
  keywords     = {General Mathematics},
  number       = {5},
  pages        = {28--36},
  publisher    = {American Mathematical Society},
  title        = {{A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II}},
  doi          = {10.1090/s1079-6762-01-00091-9},
  volume       = {7},
  year         = {2001},
}

@article{8522,
  abstract     = {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.},
  author       = {Kaloshin, Vadim and Hunt, Brian R.},
  issn         = {1079-6762},
  journal      = {Electronic Research Announcements of the American Mathematical Society},
  keywords     = {General Mathematics},
  number       = {4},
  pages        = {17--27},
  publisher    = {American Mathematical Society},
  title        = {{A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I}},
  doi          = {10.1090/s1079-6762-01-00090-7},
  volume       = {7},
  year         = {2001},
}