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