Local analytic reduction of families of diffeomorphisms

We study local analytic simplification of families of analytic maps near a hyperbolic fixed point. A particularly important application of the main result concerns families of hyperbolic saddles, where Siegelʹs theorem is too fragile, at least in the analytic category. By relaxing on the formal normal form we obtain analytic conjugacies. Since we consider families, it is more convenient to state some results for analytic maps on a Banach space; this gives no extra complications. As an example we treat a family passing through a 1 : − 1 resonant saddle.