Abstract
We consider conformal structures invariant under a volume-preserving Anosov system. We show that if such a structure is in $L^p$ for sufficiently large $p$, then it is continuous.
Highlights
In this paper we consider integrable conformal structures which are invariant under a volume preserving Anosov system
This research was motivated by recent developments in the study of rigidity properties of conformal Anosov systems
We define the Lpα,(s) spaces for conformal structures. They are a natural extension to manifolds endowed with an Anosov system of similar spaces which are standard in harmonic analysis
Summary
In this paper we consider integrable conformal structures which are invariant under a volume preserving Anosov system. The paper [dlL02] showed that conformal Anosov systems are locally differentiably rigid. More results on local and global differentiable rigidity of such systems were obtained in [Sad] and [KS03]. Various arguments were developed there to show that continuous invariant structures are differentiable. The goal of this paper is to bridge the gap between the integrable theory and the continuous/differentiable one. Once the conformal structure is known to be continuous, one can use the results of [Sad], [KS03] to obtain further regularity for the conformal structure and differential rigidity for the system. Sadovskaya structures can be made somewhat sharper than those for general solutions of cohomology equations
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