On the regularity of integrable conformal structures invariant under Anosov systems (Q1770186)

The paper deals with the volume-preserving Anosov diffeomorphism \(f: M\to M\), where \(M\) is an \(m\)-dimensional \(C^\infty\) compact manifold. Let \(E\subset TM\) be a subbundle invariant under \(Df\), \(\dim E= d\geq 2\). Roughly speaking, a conformal structure \(C_x\), \(x\in M\), is a linear operator on \(E_x\) with the norm \[ \| C_x\|= \sup_{0\neq v\in E+x} {\langle C_x v,v\rangle\over\| v\|^2}. \] With respect to these norms, the \(L^p\) spaces of the conformal structures can be defined. The authors prove that if such a structure is in \(L^p\) for sufficiently large \(p\), then it is continuous.