Local Hölder regularity of densities and Livsic theorems for non-uniformly hyperbolic diffeomor\-phisms (Q2574727)

Let \(f\) be an ergodic \(C^2\) diffeomorphism of a compact manifold \(M\) that preserves the normalized volume \(\mu\); assume that \(\mu\) has nonzero Lyapunov exponents. Consider the coboundary equation \[ \psi(x)=\chi(x)-\chi(f(x)), \] where \(\chi\) is \(\mu\)-measurable and \(\psi\) is Hölder continuous. The author shows that the solution \(\chi\) of the above equation is Hölder continuous on sets of arbitrarily large measure. This statement is a partial generalization of the known Livsic theorem from the case of Anosov diffemorphisms to the case of nonuniformly hyperbolic diffeomorphisms. The main application of this result is to densities of absolutely continuous measures. Let \(\mu\) be an \(f\)-invariant ergodic probability measure that is equivalent to the volume, i.e., \(d\mu=\rho d\omega\), where \(\omega\) is the volume form on \(M\) and \(\rho\) is the density of the measure \(\mu\). It is shown that if \(\mu\) has nonzero Lyapunov exponents, then the density \(\rho\) is Hölder continuous on sets of arbitrarily large measure.