Invariant Manifolds for Non-differentiable Operators

2017/04/20 by Marco Martens, Martens, M., Liviana Palmisano +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1704.06328

openalex publication_date 2017/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A general invariant manifold theorem is needed to study the topological classes of smooth dynamical systems. These classes are often invariant under renormalization. The classical invariant manifold theorem cannot be applied, because the renormalization operator for smooth systems is not differentiable and sometimes does not have an attractor. Examples are the renormalization operator for general smooth dynamics, such as unimodal dynamics, circle dynamics, Cherry dynamics, Lorenz dynamics, Hénon dynamics, etc. A general method to construct invariant manifolds of non-differentiable non-linear operators is presented. An application is that the \mathcal C^4+ε Fibonacci Cherry maps form a \mathcal C¹ codimension one manifold.

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