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Let $M$ be a compact finite-dimensional manifold and $f\colon M\to M$ be a diffeomrphism. By $P_n(f)$ we denote the number of periodic points of $f$ with period $n$, that is, the number of fixed points of $f^n$. Katok Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publications Mathématiques de l'IHÉS 51, no. 1 (1980), pp. 137-173 showed that the topological entropy of a $\mathcal{C}^{1+\varepsilon}$ diffeomorphism of a compact surface obeys the following inequality: $$ \limsup_{n\to\infty} \frac{\log P_n(f)}{n}\ge h(f). $$ He also conjectured (see paragraph 5 page 141 of the article cited above) that this inequality holds in any dimension generically in the $\mathcal{C}^1$-topology. In other words: There is a dense $G_\delta$ set $\mathcal{G} \subset \text{Diff}^1 (M)$ such that for any $f\in\mathcal{G}$ one has $$ \limsup_{n\to\infty} \frac{\log P_n(f)}{n}\ge h(f). $$ Is this conjecture still open? I would be grateful for any references.

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Nice question. I think it has not been directly adressed in the literature, but combining known results it seems that a positive answer to that question can be given. Still some details must be carried out (which I did not) to be sure.

The main point is that $C^1$-far from homoclinic tangencies the result must be true due to the existence of a maximal entropy measure (see here, here, here and references therein) where the splitting has one dimensional center (and so Katok type results can be carried out, see here). There is a subtelty here which is that the central exponent may be a priori equal to zero, in which case one would like to use the genericity hypothesis to show that the estimate holds (I think the concept of principal simbolic extension in the above references should be helpful for this). Here there might be some work to be done.

In the open set where homoclinic tangencies are dense, results of Kaloshin and from Bonatti-Diaz-Fisher suggest that there should be superexponential growth of periodic orbits, but some care must be taken since the precise statements do not work for this case. On the other hand, since what is needed is way less than superexponential growth, I don't see this part as being the hard one.

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