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

TL;DR

The paper proves that any diffeomorphism on a manifold of the form rac{rac{1}{z} imes Mrac{rac{1}{z}} can be realized as a total renormalization of a map arbitrarily close to the identity, allowing localization of complex dynamical properties.

Contribution

It introduces a method to represent any diffeomorphism as a total renormalization of a near-identity map on certain manifolds.

Findings

Any diffeomorphism can be realized as a total renormalization of a close-to-identity map.

Enables localization of complex dynamical properties near the identity.

Facilitates the construction of systems with properties like Bernoulli behavior.

Abstract

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.