Let \(M\) be a smooth compact finite-dimensional manifold, and let \(f\) be a \(C^{r}\)-diffeomorphism on \(M\) for some \(r\in [0,\infty ]\). Denote by \(P_{n}(f)\) the number of periodic points of \(f\) having period \(n\). The growth rate of \(P_{n}(f)\) gives an information on the “chaoticity” (or “complexity”) of the map \(f\).
According to a result obtained in M. Artin and B. Mazur [Ann. Math. 81, 82–99 (1965; Zbl 0127.13401)], diffeomorphisms satisfying \(P_{n}(f)\leq\exp (Cn)\) for some \(C\) are dense in the set of all \(C^{r}\)-diffeomorphisms endowed with the uniform \(C^{r}\)-topology. On the other hand for \(r\geq 2\) it has been proved in [the first author, Commun. Math. Phys. 211, No. 1, 253–271 (2000; Zbl 0956.37017)] that there is a Baire generic set of \(C^{r}\)-diffeomorphisms having arbitrarily large growth rate of \(P_{n}(f)\). The question addressed in this paper is how does the growth rate of \(P_{n}(f)\) behave for the “most” diffeomorphisms from a measure theoretic point of view. To this end the notion of prevalence is used in order to describe the meaning of “a \(C^{r}\)-diffeomorphism satisfies a certain property with probability one”.
The manifold \(M\) is embedded in the interior of the closed unit ball \(B^{N}\subset{\mathbb R}^{N}\) for a sufficiently large \(N\). Then \(f\) is extended to an element of \(C^{r}(B^{N},{\mathbb R}^{N})\) in a suitable way. For a “small” function \(g\in C^{r}(B^{N},{\mathbb R}^{N})\) the map \(f+g\) corresponds in a natural way to a \(C^{r}\)-diffeomorphism \(h\) on \(M\). A property is called prevalent, if it holds on a Borel set \(S\subset C^{r}(B^{N},{\mathbb R}^{N})\) and there exists a Borel probability measure \(\mu\) on \(C^{r}(B^{N},{\mathbb R}^{N})\) with compact support such that for all \(f\in C^{r}(B^{N},{\mathbb R}^{N})\) there exists a Borel set \(S_{f}\subset C^{r}(B^{N},{\mathbb R}^{N})\) with \(\mu (S_{f})=1\) and \(f+g\in S\) for all \(g\in S_{f}\).
For a linear map \(L:{\mathbb C}^{N}\to{\mathbb C}^{N}\) the hyperbolicity \(\gamma (L)\) is defined as the infimum of \(| Lv-zv| \), where the infimum is taken over all \(v\in{\mathbb C}^{N}\) with \(| v| =1\) and all \(z\in{\mathbb C}\) with \(| z| =1\). Define \(\gamma_{n}(f):=\inf\gamma (df^{n}(x))\), where the infimum is taken over all periodic points \(x\) of period \(n\). One calls the decay of \(\gamma_{n}(f)\) the decay of hyperbolicity of periodic points of \(f\). The main result of this paper states that for \(1<r\leq\infty\), a prevalent \(C^{r}\)-diffeomorphism \(f\) and \(\delta >0\) there exists a constant \(C\) such that \(P_{n}(f)\leq\exp (Cn^{1+\delta })\) and \(\gamma_{n}(f)\geq\exp (-Cn^{1+ \delta })\).
Before giving detailled proofs the authors describe the main ideas of the proof. This is very helpful for readers. One main method in the proof is the use of Newton interpolation polynomials in order to control orbits of perturbed maps. Then a detailled proof of the result in the one-dimensional case is given. Finally some ideas how to generalize the proof for the more-dimensional case are presented. Again it is very helpful for readers to see at first the proof of the one-dimensional case, because the ideas are much better visible than in the more-dimensional case. The proof of the more-dimensional case itself is deferred to a forth-coming paper.
Although the result and the proof are nice in the one-dimensional case, it has to be mentioned that there exists a better result for \(C^{2}\)-maps on the interval. In fact, it is also mentioned by the authors that it has been proved in [M. Martens, W. de Melo and S. van Strien, Acta Math. 168, No. 3–4, 273–318 (1992; Zbl 0761.58007)] that for every \(C^{2}\)-map on the interval without flat critical points there is a \(\gamma >0\) such that \(\gamma_{n}(f)\geq\gamma\) for all sufficiently large \(n\). This implies also that \(P_{n}(f)\leq\exp (Cn)\) for some \(C\).