Regularity of Non-stationary Stable Foliations of Toral Anosov Maps

Abstract

We consider a sequence of $C^2$ (or $C^3$) Anosov maps of the two-dimensional torus that satisfy a common cone condition, and show that if their $C^2$ (respectively, $C^3$) norms are uniformly bounded, then the non-stationary stable foliation must be of class $C^1$ (respectively, $C^{1+\text{H\"older}}$). This generalizes the classical results on smoothness of the invariant foliations of Anosov maps. We also provide an example that shows that an assumption on boundedness of the norms cannot be removed, which is a phenomenon that does not have an analog in the stationary setting. The main motivation stems from a standing conjecture concerning the dimension properties of the spectra of Sturmian Hamiltonian operators, and this result serves as a first step towards addressing this conjecture. A detailed appendix is provided showing the potential argument and connection between this theory of non-stationary hyperbolic dynamics and the spectral dimension of these operators. We also provide an addendum demonstrating that a similar result holds for a sequence of Anosov maps of the $d$-dimensional torus whose stable directions have codimension $1$.