Flow-independent Anisotropic space, and perturbation of resonances

Given an Anosov vector field $X_0$, all sufficiently close vector fields also are of Anosov type. In this note, we check that one can choose the so-called "Faure-Sj\"ostrand" anistropic spaces to be adapted to any smooth vector field sufficiently close to $X_0$ in $C^1$ norm. It follows that finite families of resonances depend smoothly on the vector field with the usual caveats. In this second version, more details are given on the perturbation of spectral projectors.