We show that for any $ 1 \le r \le \infty $, any $C^r$-diffeomorphism in the connected component of the identity, on a manifold of the form $ M \times \T $ with $ \dim M \ge 1 $, is conjugated to the first return of an arbitrarily close to identity map on a domain $ \Delta \subset M $. Moreover the renormalization domain $ \Delta$ can be chosen such that its orbit cover the whole manifold $ M \times \T$. This enables us to localize nearby the identity the existence of many properties in dynamical systems, such as being Bernoulli for a smooth volume form or universal with renormalization domains of arbitrarily large mass. This is a joint work with Pierre Berger and Nicolas Gourmelon.