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

2022/10/17 by Pierre Berger, Berger, Pierre, Nicolaz Gourmelon +3 · 2 citations
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2210.09064

Abstract

For any 1≤ r≤ ∞, we show that every diffeomorphism of a manifold of the form ℝ/ℤ × M is a total renormalization of a C^r-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.

Cited by

• Wild dynamics on manifolds
• On Kolmogorov-typical properties of symplectic dynamics

Related

• Realizing arbitrary d-dimensional dynamics by renormalization of C^d-perturbations of identity
• Maps close to identity and universal maps in the Newhouse domain
• Renormalization of Unicritical Diffeomorphisms of the Disk
• The full renormalization horseshoe for multimodal maps
• A hyperbolic diffeomorphism with countably many ergodic components near identity