Joint work with Nicolaz Gourmelon and Mathieu Helfter.

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 nearby the identity the existence of many properties in dynamical systems, such as being Bernoulli for a smooth volume form.