Addendum to: Symbolic dynamics for the $N$-centre problem at negative energies

TL;DR

This paper clarifies a technical step in the proof of a theorem related to symbolic dynamics in the N-centre problem at negative energies, focusing on the differentiability of a key function.

Contribution

It completes and clarifies a crucial part of the proof in the authors' previous work, ensuring the differentiability assumption holds.

Findings

Confirmed the differentiability of the function F used in the proof.

Resolved a delicate step in the symbolic dynamics analysis.

Strengthened the theoretical foundation of the N-centre problem at negative energies.

Abstract

This paper aims at completing and clarifying a delicate step in the proof of Theorem 5.3 of our paper \cite{ST}, where it was used the differentiability of a function $F$, which a priori can appear not necessarily differentiable.