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

For any $1\le r\le \mathrm{\infty }$, we show that every diffeomorphism of a manifold of the form $\mathbb{R}\mathrm{/}\mathbb{Z}\times M$ is a total renormalization of a $C^r$-close to identity map. In other words, for every diffeomorphism $f$ of $\mathbb{R}\mathrm{/}\mathbb{Z}\times M$, 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 $\mathbb{R}\mathrm{/}\mathbb{Z}\times M$. 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.