[Paper Review] An Isotopic Perturbation Lemma Along Periodic Orbits

Nicolas Gourmelon|arXiv (Cornell University)|Dec 29, 2012

Mathematical Dynamics and Fractals10 references3 citations

TL;DR

This paper establishes a refined version of Franks' isotopic perturbation lemma for $C^1$-diffeomorphisms, showing that if a path of linear cocycles along a periodic orbit preserves strong stable/unstable directions of fixed dimension, then the perturbation can be made to preserve the corresponding semi-local stable and unstable manifolds. The key contribution is a pathwise isotopic perturbation technique that preserves invariant manifolds under $C^1$-perturbations, enabling control over homoclinic structures in $C^1$-generic dynamics.

ABSTRACT

A well-known lemma by John Franks asserts that one obtains any perturbation of the derivative of a diffeomorphism along a periodic orbit by a $C^1$-perturbation of the whole diffeomorphism on a small neighbourhood of the orbit. However, one does not control where the invariant manifolds of the orbit are, after perturbation. We show that if the perturbated derivative is obtained by an isotopy along which some strong stable/unstable manifolds of some dimensions exist, then the Franks perturbation can be done preserving the corresponding stable/unstable semi-local manifolds. This is a general perturbative tool in $C^1$-dynamics that has many consequences. We give simple examples of such consequences, for instance a generic dichotomy between dominated splitting and small stable/unstable angles inside homoclinic classes.

Motivation & Objective

• To address the lack of control over invariant manifolds after $C^1$-perturbations in Franks' original lemma.
• To develop a method that preserves semi-local stable and unstable manifolds during perturbations along periodic orbits.
• To provide a systematic tool for constructing homoclinic tangencies and heterodimensional cycles within homoclinic classes lacking strong domination.
• To generalize perturbation techniques to preserve entire flags of strong manifolds under isotopic paths of linear cocycles.
• To enable applications in $C^1$-generic dynamics, including dichotomies between dominated splittings and small angles in homoclinic classes.

Proposed method

• The method uses isotopies of linear cocycles along periodic orbits, ensuring that strong stable/unstable directions of fixed dimension persist throughout the path.
• It constructs a $C^1$-perturbation of the diffeomorphism that matches a prescribed path of derivatives along the orbit, while preserving the semi-local structure of invariant manifolds.
• The construction relies on identifying admissible flags of invariant manifolds and ensuring the perturbation preserves them within annuli of fundamental domains.
• The key technical condition is that the $i$-strong stable eigenvalues at some time $t_0$ have equal moduli, enabling the preservation of $j$-strong stable manifolds for all $j \leq i$.
• The perturbation is localized to a neighborhood of the periodic orbit and agrees with the original diffeomorphism outside this region.
• The method extends to conservative and symplectic settings, with technical adaptations required for codimension-one manifolds.

Experimental results

Research questions

• RQ1 Can $C^1$-perturbations along periodic orbits preserve the structure of strong stable and unstable manifolds?
• RQ2 Under what conditions on the path of linear cocycles can invariant manifolds be preserved during perturbations?
• RQ3 How can one systematically create homoclinic tangencies or heterodimensional cycles within homoclinic classes lacking dominated splittings?
• RQ4 Can the Franks lemma be generalized to preserve entire flags of strong manifolds via isotopic paths of derivatives?
• RQ5 What are the implications of this preservation for $C^1$-generic dynamics, particularly in relation to dichotomies between domination and small angles?

Key findings

• The perturbation preserves the $j$-strong stable manifolds for all $j \leq i$ within an annulus of the $i$-strong stable manifold, provided the $i$ strongest eigenvalues at some $t_0$ have equal moduli.
• The size of the perturbation can be made arbitrarily small, proportional to the size of the path of matrices $({\mathcal{A}}_t)$.
• The perturbation preserves the orbit and agrees with the original diffeomorphism outside a small neighborhood of the periodic orbit.
• The method allows for the construction of homoclinic tangencies within homoclinic classes where no uniform dominated splitting exists.
• The result implies a generic dichotomy between dominated splittings and small stable/unstable angles in homoclinic classes.
• The technique extends to conservative and symplectic settings, though codimension-one manifolds present technical obstructions to semi-local preservation.