Local perturbations of conservative $C^1$-diffeomorphisms

TL;DR

This paper extends techniques for perturbing $C^1$-diffeomorphisms, especially in volume-preserving and symplectic systems, to better understand their entropy and invariant sets without relying on genericity.

Contribution

It generalizes local perturbation methods from dissipative to volume-preserving and symplectic $C^1$-diffeomorphisms, enabling new insights into their entropy and invariant set approximation.

Findings

Extended perturbation techniques to symplectic systems.

Applied methods to study entropy of $C^1$-diffeomorphisms.

Provided a way to approximate transitive invariant sets without genericity.

Abstract

A number of techniques have been developed to perturb the dynamics of $C^1$-diffeomorphisms and to modify the properties of their periodic orbits. For instance, one can locally linearize the dynamics, change the tangent dynamics, or create local homoclinic orbits. These techniques have been crucial for the understanding of $C^1$ dynamics, but their most precise forms have mostly been shown in the dissipative setting. This work extends these results to volume-preserving and especially symplectic systems. These tools underlie our study of the entropy of $C^1$-diffeomorphisms in (arxiv:1606.01765). We also give an application to the approximation of transitive invariant sets without genericity assumptions.