Regularity of the conjugacy in the real-spectrum, non-constant periodic data case on T^3

Determine the optimal regularity of the conjugating homeomorphism between C^r Anosov diffeomorphisms on the 3-torus that are C^0-conjugate and have matching periodic data when the associated hyperbolic toral automorphism has a real spectrum (i.e., no complex conjugate pair) and the matching periodic data are non-constant; in particular, ascertain whether higher regularity such as C^{1+Hölder} up to smooth conjugacy must hold in this real-spectrum setting.

Background

The matching periodic data problem asks when a C^0 conjugacy between Anosov diffeomorphisms upgrades to higher regularity under the assumption that derivatives at periodic points match up to conjugacy. On T^3, substantial progress has been made in the local (near a hyperbolic toral automorphism) setting. When the unstable spectrum has a complex conjugate pair, existing techniques yield smooth conjugacy under matching periodic data; in contrast, when the spectrum is real, current results provide C^{1+Hölder} conjugacy locally but not smoothness.

The present paper resolves the remaining local case in the complex-spectrum setting, showing smoothness under non-constant matching periodic data together with an SRB–MME coincidence assumption. The authors emphasize that their methods rely crucially on the presence of complex eigenvalues, and they point to an explicit open problem (GRH, Problem 1.6) concerning the regularity of conjugacy in the real-spectrum, non-constant periodic data case, which remains unresolved.