Finite-Regularity Counterexamples to Katok's Periodic-Point Conjecture

For any $r>1$ and any nonnegative sequence $ω(n)=o(n)$, we construct a $C^r$ sphere diffeomorphism $f$ with $h=h_{\mathrm{top}}(f)>0$ such that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\log\#\operatorname{Fix}(f^n)=h$ and $\displaystyle \sum_{n\geqslant 1}e^{-hn+ω(n)}\#\operatorname{Fix}(f^n) < \infty$. Every ergodic invariant probability measure is hyperbolic, and, apart from finitely many fixed points, all periodic points lie in a single topologically mixing homoclinic class of entropy $h$. This shows that the logarithmic periodic-point lower bound is optimal in finite regularity. In particular, we answer the finite-regularity case of Katok's periodic-point question in the negative.ative.