Correction to: Infinitesimal Lyapunov functions for singular flows

1 Correction to: Math. Z. (2013) 275:863–897 https://doi.org/10.1007/s00209-013-1163-8

• On p. 874: In the statement of Lemma 2.2, which reads “Let V be a real finite dimensional vector space endowed with a non-positive definite and non-degenerate quadratic form”, should read “Let V be a real finite dimensional vector space endowed with a indefinite non-degenerate quadratic form”.

• On p. 875 at item (2) of Theorem 2.7, the last line should be replaced by:

In particular, we get $\partial _t\log |\mathcal {J}(A_t(x)v)|\ge \delta (X_t(x)), x\in V, t\ge 0, v\in E_x$ and $\mathcal {J}(v)>0$; or $\partial _t\log |\mathcal {J}(A_t(x)v)|\le \delta (X_t(x)), x\in V, t\ge 0, v\in E_x$ and $\mathcal {J}(v)<0$.

• On p. 876: although not used anywhere in the article, item (5) of Theorem 2.7 is false. The proof presented in p. 878 has a wrong sign in the calculation. The corrected calculation gives only $\partial _t\left( \frac{|\mathcal {J}(A_t(x)w)|}{\mathcal {J}(A_t(x)v)}\right) \mid _{t=0}\ge 0$ for the $\mathcal {J}$-separated cocycle $A_t(x)$ and strictly positive if $A_t(x)$ is a strictly $\mathcal {J}$-separated cocycle.

• On p. 884: items (1) and (2) in the statement of Theorem 2.23 in [1] need to be corrected, since [1, Example 5 with index 2] shows that hyperbolic behavior of the subbundles cannot be expressed as necessary and sufficient conditions using the sign of the function $\delta$. Item (3) of [1, Thm. 2.7] is correct. We reformulate the statement of items (1) and (2) as follows.

Theorem 0.1

[1, Theorem 2.23] Let $\Gamma$ be a compact invariant set for $X_t$ admitting a dominated splitting $E_\Gamma = F_-\oplus F_+$ for a linear multiplicative cocycle $A_t(x)$ over $\Gamma$ with values in E. Let $\mathcal {J}$ be a $C^1$ field of indefinite quadratic forms such that $A_t(x)$ is strictly $\mathcal {J}$-separated admitting a function $\delta :\Gamma \rightarrow {\mathbb R}$ as given in Theorem 2.7. Then

• (1) If $\Delta _s^t(x)\xrightarrow [(t-s)\rightarrow +\infty ]{}-\infty$ for all $x\in \Gamma$, then $F_-$ is a uniformly contracted subbundle.

• (2) If $\Delta _s^t(x)\xrightarrow [(t-s)\rightarrow +\infty ]{} +\infty$ for all $x\in \Gamma$, then $F_+$ is a uniformly expanding subbundle.

Since the short proof of these items uses the corrected expressions in item (2) of Theorem 2.7, we present it below.

Proof

If $\Delta _0^t(x)\xrightarrow [t\rightarrow +\infty ]{}-\infty$, then from item (2) of [1, Thm 2.7] we get $\frac{\mathcal {J}(A_t(x)v)}{\mathcal {J}(v)}\le e^{\Delta _0^t(x)}\xrightarrow [t\rightarrow +\infty ]{}0$ for all $x\in \Gamma$ and $v\in F_-(x)$. So $F_-$ in uniformly contracted, by [1, Lemmas 2.18 & 2.24].

If $\Delta _s^t(x)\xrightarrow [(t-s)\rightarrow +\infty ]{} +\infty$, then analogously $\frac{\mathcal {J}(A_t(x)v)}{\mathcal {J}(v)}\ge e^{\Delta _0^t(x)}\xrightarrow [t\rightarrow +\infty ]{}+\infty$ for all $x\in \Gamma$ and $v\in F_+(x)$. So $F_+$ in uniformly contracted, again by [1, Lemmas 2.18 & 2.24]. $\square$