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

For any $1\le r\le \mathrm{\infty }$ $1\le r\le \infty $ , we show that every diffeomorphism of a manifold of the form $\mathbb{R}/\mathbb{Z}\times M$ $\mathbb{R}/\mathbb{Z}\times M$ is a total renormalization of a $C^r$ $C^{r}$ -close to identity map. In other words, for every diffeomorphism $f$ $f$ of $\mathbb{R}/\mathbb{Z}\times M$ $\mathbb{R}/\mathbb{Z}\times M$ , there exists a map $g$ $g$ arbitrarily close to identity such that the first return map of $g$ $g$ to a domain is conjugate to $f$ $f$ and moreover the orbit of this domain is equal to $\mathbb{R}/\mathbb{Z}\times M$ $\mathbb{R}/\mathbb{Z}\times M$ . 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.