Correction to: Exponentially Small Splitting of Separatrices Associated to 3D Whiskered Tori with Cubic Frequencies

Correction to: Commun. Math. Phys. 378, 1931–1976 (2020) https://doi.org/10.1007/s00220-020-03832-y

We correct an error in Sect. 2.1 of [DGG20]. Just before Proposition 4, it is claimed that one can choose a matrix $T=T(\omega )\in \textrm{SL}(3,\mathbb Z)$ whose real eigenvalue $\lambda =\lambda (\omega )>1$ is minimal or, equivalently, the norm $\left| T\right|$ is minimal, and this matrix T is called “the principal Koch’s matrix for $\omega$ ” (we denote $\left| \cdot \right| =\left| \cdot \right| _2$ the matrix norm subordinate to the Euclidean norm for vectors). This equivalence is not true, and the matrix $T(\omega )$ should be defined simply as the one whose real eigenvalue $\lambda =\lambda (\omega )>1$ is minimal.

The following example shows that the principal Koch’s matrix is not necessarily the Koch’s matrix whose norm is minimal. Let us consider the frequency vector $\omega =(1,\Omega ,\Omega ^2)$ where $\Omega$ is the cubic golden number: the real root of $\Omega ^3=1-\Omega$ (see Sect. 2.3). The principal Koch matrix for this vector is $\displaystyle T(\omega )=\left( \begin{array}{ccc} 1 &{}0 &{}1\\ 1 &{}0 &{}0\\ 0 &{}1 &{}0 \end{array}\right)$ with the real eigenvalue $\lambda =1+\Omega ^2=1/\Omega$ (see (68)). Now consider the vector $\widetilde{\omega }=(1,\,\Omega ^2,\,-1+\Omega +\Omega ^2)$. We have the relation $\widetilde{\omega }=S\omega$, where the matrix $\displaystyle S=\left( \begin{array}{ccc} 1 &{}0 &{}0\\ 0 &{}0 &{}1\\ -1 &{}1 &{}1 \end{array}\right)$ is unimodular. It is clear that the principal Koch’s matrix for this new vector is $\displaystyle T(\widetilde{\omega })=S\,T(\omega )\,S^{-1}=\left( \begin{array}{ccc} 1 &{}1 &{}0\\ 1 &{}-1 &{}1\\ 1 &{}-2 &{}1 \end{array}\right)$, with the same eigenvalue $\lambda (\widetilde{\omega })=\lambda (\omega )$. But its norm $\left| T(\widetilde{\omega })\right| \approx 2.978400$ is not the minimal one, since $T(\widetilde{\omega })^2$ is also a Koch’s matrix, with smaller norm $\left| T(\widetilde{\omega })^2\right| \approx 2.457837$ .

Hence, the paragraph previous to Proposition 4 should be rewritten as follows:

Using this lemma, we next show the “uniqueness” of the matrix T satisfying Koch’s result. More precisely, we can choose $T=T(\omega )\in \textrm{SL}(3,\mathbb Z)$ whose real eigenvalue $\lambda =\lambda (\omega )>1$ is minimal, and we call this matrix “the principal Koch’s matrix for $\omega$ ”. This matrix T is not necessarily the one of minimal norm among the Koch’s matrices for $\omega$.

Although the validity of the results of the paper is not affected by this mistake, the algorithm for determining the principal Koch’s matrix in a concrete case, described between Lemma 5 and Remark 6, is no longer valid since it relies in finding the Koch’s matrix of minimal norm $\left| T\right|$. Alternatively, to find the Koch’s matrix with minimal real eigenvalue $\lambda$, we should reformulate this algorithm in the following way:

Now, in order to determine the principal Koch’s matrix for $\omega$ we can carry out the following simple exploration. We consider the (integer) entries of the first row $T_{(1)}$ as successive data, say with increasing norm $\left| T_{(1)}\right|$, until the whole matrix T determined from Lemma 5 belongs to $\textrm{SL}(3,\mathbb Z)$ (i.e. it has integer entries and determinant 1) and has an eigenvalue $\lambda >1$ in (30). By Koch’s result, we know that such a matrix exists and will be reached after a finite exploration. It remains to check whether the matrix $T^*$, with eigenvalue $\lambda ^*$, obtained in this way is the principal Koch’s matrix for $\omega$ since, in principle, there could exist another Koch’s matrix T with $\left| T_{(1)}\right| \ge \left| T^*_{(1)}\right|$ but with eigenvalue $\lambda <\lambda ^*$. If this happens, such a new matrix T would satisfy $\left| T_{(1)}\right|<\left| T\right| \le \kappa \lambda <\kappa \lambda ^*$, where $\kappa =\kappa (\omega )$ is the condition number introduced in (33), and the inequality $\left| T\right| \le \kappa \lambda$ appears in the proof of Lemma 3. Hence, after obtaining a first matrix $T^*$, it is enough to continue the exploration with increasing norms $\left| T_{(1)}\right|$ up to the value $\kappa \lambda ^*$ and, if a new Koch’s matrix T is obtained, check if its real eigenvalue $\lambda$ is lower than $\lambda ^*$, which would imply that the matrix T has to replace $T^*$ as the principal one.