Periodic growth for analytic representatives

Construct, in every pseudo-Anosov isotopy class, a real-analytic diffeomorphism g with finitely many fixed points for every iterate and with exponential periodic-point growth strictly greater than its topological entropy.

Background

The paper shows that smooth representatives can have periodic-point growth exceeding topological entropy, even with finite fixed-point sets for every iterate, using perturbations associated with homoclinic tangencies and Kaloshin’s theorem. It leaves open whether an analogous phenomenon can be realized by a real-analytic representative in every pseudo-Anosov isotopy class.

References

Does every pseudo-Anosov isotopy class contain a real-analytic diffeomorphism g, with #\operatorname{Fix}(gn)<\infty for every n, whose exponential periodic-point growth rate exceeds h_{\mathrm{top}(g)? Namely, (g)<\limsup_{n\to\infty}\frac1n\log\max{1,#\operatorname{Fix}(gn)}?

— Entropy and periodic points in pseudo-Anosov isotopy classes  (2610.26230 - Carrasco et al., 19 Oct 2026) in Section 6, question titled “Periodic growth for analytic maps”