Regularity of Non-stationary Stable Foliations of Toral Anosov Maps

TL;DR

The paper proves that for a non-stationary sequence of $C^2$ (resp. $C^3$) Anosov diffeomorphisms on the 2-torus satisfying a common cone condition, uniform bounds on the $C^2$ (resp. $C^3$) norms of the maps and their inverses imply a $C^1$ (resp. $C^{1+\beta}$) regularity of the non-stationary stable foliation, with the stable direction field being $C^1$ (resp. $C^{1+\beta}$). The core method is a non-stationary version of the $C^r$ section theorem, implemented via a graph-transform on a skew-product, yielding a $C^1$ (or $C^{1+\beta}$) stable direction and foliation. A key contribution is the demonstration that the uniform norm-boundedness assumption is essential, via an explicit counterexample, and the work extends to the codimension-1 stable direction case as an addendum. These results provide a foundational step toward understanding spectral dimensions in Sturmian Hamiltonians through non-stationary hyperbolic dynamics and trace-map dynamics.

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, $C^{1+\text{Hölder}}$). 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 $1$.

Figures (4)

• Figure 1: Image of the cone $K^s(x)$ under $D_xf^{-1}$.
• Figure 2: Illustration of Theorem \ref{['Torus Section Theorem']}
• Figure 3: Illustration of rectangles $R_n$ and $R_n'$. The blue lines represent the boundaries of the unstable cones $K^u(p_n)\cap R_n$ and $K^u\left(p_n'\right)\cap R_n'$. The leaves of $\mathcal{F}$ connecting $\kappa$ and $H(\kappa)$ as in property (P2) must stay between the dashed lines, since $\kappa$ would remain in the left cone, which guarantees that property (P2) holds.
• Figure 4: Illustration of $f_N\circ\cdots \circ f_1(I)$ and $H_N\left(f_N\circ\cdots \circ f_1(I)\right)$. These curves remain within the dashed blue lines since they are tangent to $K^u$. The leaves from $f_N\circ \cdots \circ f_1\left(W^s\left(\overline f \right)\right)$ connecting these curves remain within the red dashed lines since they are tangent to $K^s$. Using the geometry of these cones, one can establish (\ref{['comparing transversal lengths']}).

Theorems & Definitions (29)

• Theorem 1
• Conjecture 1
• Lemma 1
• proof
• Theorem 2
• Remark 1
• Remark 2
• Remark 3
• Theorem 3
• proof