Every diffeomorphism is a total renormalization of a close to identity map

For any 1 ≤ r ≤ ∞, we show that every diffeomorphism of a manifold of the form R/Z × M is a total renormalization of a Cr-close to identity map. In other words, for every diffeomorphism f of R/Z×M, there exists a map g arbitrarily close to identity such that the first return map of g to a domain is conjugate to f and moreover the orbit of this domain is equal to R/Z×M. This enables us to localize near the identity the existence of many properties in dynamical systems, such as being Bernoulli for a smooth volume form.