[Paper Review] Invariant Manifolds for Non-differentiable Operators

Marco Martens, Liviana Palmisano|arXiv (Cornell University)|Apr 20, 2017

Mathematical Dynamics and Fractals29 references4 citations

TL;DR

This paper establishes a general invariant manifold theorem for non-differentiable nonlinear operators, replacing classical differentiability and hyperbolicity conditions with 'jump-out differentiability' and topological hyperbolicity. It proves that the space of $\mathcal{C}^{4+\epsilon}$ Fibonacci Cherry maps forms a $\mathcal{C}^1$ codimension one manifold, extending the applicability of invariant manifold theory to non-smooth renormalization schemes in dynamics.

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 $\mathcal C^1$ codimension one manifold.

Motivation & Objective

• To develop a general invariant manifold theorem applicable to non-differentiable operators in dynamical systems, particularly where classical theorems fail.
• To address the lack of differentiability and attractors in renormalization operators for smooth systems like unimodal, Cherry, Lorenz, and Hénon dynamics.
• To prove that topological classes in smooth dynamics—specifically Fibonacci Cherry maps—are smooth manifolds despite non-differentiable renormalization.
• To establish a framework for analyzing geometric rigidity of conjugacies between systems with the same topology.

Proposed method

• Introduce 'jump-out differentiability' as a weaker substitute for classical differentiability, enabling derivative-like control in non-smooth settings.
• Use curve dynamics with Lipschitz regularity instead of differentiability to analyze the graph transform method on decomposed systems.
• Define a decomposition of systems into components $\varphi^r$, $\varphi^l$ to make renormalization jump-out differentiable.
• Construct the invariant manifold via the graph transform method, relying on uniform boundedness of partial derivatives $D_{rs}$ and $D_{rl}$ on $X_2$.
• Establish continuity and H"older-type estimates for the derivative operators: $ \left| \tilde{\varphi}^r(\underline{f}+\Delta\underline{\varphi}) - [\tilde{\varphi}^r(\underline{f}) + D\Delta\underline{\varphi}] \right|_2 = O_{\underline{f}}(|\Delta\underline{\varphi}|_2^{1+\epsilon}) $.
• Prove that the partial derivatives $ \frac{\partial\tilde{\varphi}^r_\tau}{\partial\varphi_{\tau'}} $ extend to uniformly bounded operators $ D_{rs}, D_{rl} : X_2 \to X_2 $.

Experimental results

Research questions

• RQ1 Can invariant manifolds be constructed for non-differentiable renormalization operators in smooth dynamical systems?
• RQ2 Is the space of $\mathcal{C}^{4+\epsilon}$ Fibonacci Cherry maps a smooth manifold despite non-differentiable renormalization?
• RQ3 Does topological conjugacy between systems imply geometric rigidity, i.e., $\mathcal{C}^{1+\beta}$ regularity of the conjugacy?
• RQ4 Can the classical invariant manifold theorem be generalized beyond differentiable and hyperbolic settings?
• RQ5 What quantitative conditions ensure the existence of invariant manifolds when attractors do not exist?

Key findings

• The $\mathcal{C}^{4+\epsilon}$ Fibonacci Cherry maps form a $\mathcal{C}^1$ codimension one manifold, as stated in Theorem 6.1.
• The renormalization operator for Fibonacci Cherry dynamics is not differentiable and lacks an attractor, yet the invariant manifold still exists.
• The partial derivatives $ \frac{\partial\tilde{\varphi}^r_\tau}{\partial\varphi_{\tau'}} $ are uniformly bounded and extend to bounded operators $ D_{rs}, D_{rl} $ on $ X_2 $.
• The dependence of the renormalization map on perturbations satisfies a H"older-type estimate: $ \left| \tilde{\varphi}^r(\underline{f}+\Delta\underline{\varphi}) - [\tilde{\varphi}^r(\underline{f}) + D\Delta\underline{\varphi}] \right|_2 = O_{\underline{f}}(|\Delta\underline{\varphi}|_2^{1+\epsilon}) $.
• The method applies broadly to smooth one-dimensional dynamics, Hénon dynamics, and other systems where classical theorems fail due to non-differentiability.