Regularity of invariant graphs over hyperbolic systems

Abstract

We consider cocycles with negative Lyapunov exponents defined over a hyperbolic dynamical system. It is well known that such systems possess invariant graphs and that under spectral assumptions these graphs have some degree of Hölder regularity. When the invariant graph has a slightly higher Hölder exponent than the a priori lower bound on an open set (even on just a set of positive measure for certain systems), we show that the graph must be Lipschitz or (in the Anosov case) as smooth as the cocycle.

Hyperbolic graphs: Critical regularity and box dimension

“…Remark 1.6. The fact that Φ is either Lipschitz or has a maximal Hölder exponent is often referred to as critical regularity and has already been proven in our setting in [15]. We reproduce this result here both for the convenience of the reader and due to the fact that this will be a byproduct of the methods for computing the box dimension, and we have to introduce the respective concepts and estimates anyway.…”

“…To prove the above proposition, we follow very closely and extend [15], in particular since proofs there are given in the particular case of τ Anosov.…”

“…higher-dimensional) systems. We proceed by studying graphs in three-dimensional skew product systems T : Ξ × R → Ξ × R, T (ξ, x) = (τ (ξ), T ξ (x)), with hyperbolic surface diffeomorphisms, or their restrictions to basic pieces τ : Ξ → Ξ, in the base, building on previous results in [22,3,15]. Summarizing our main results, except in a nongeneric case when the graph is Lipschitz, its box dimension is given by d s + d, where d s is the dimension of stable slices of Ξ and where d is determined as the unique solution of the pressure equation…”

The box dimension of invariant graphs over hyperbolic systems

“…By [3], we know that if is Lipschitz on any open set then it must be globally Lipschitz (and in particular smooth and have box dimension over a stable manifold equal to the box dimension of the stable manifold itself). We are assuming that is not Lipschitz on any open set.…”

“…Moreover, this graph is always Ho¨lder. In [3] the higher regularity of was discussed. The graph is always as smooth as g along unstable manifolds, but along stable manifolds the following dichotomy holds: there exists a critical Ho¨lder exponent g crit 2 ð0, 1Þ such that either:…”