$\mu$-a.e. $x\in X$, a splitting ${\mathbb{R}}^2 = E^u(x) \oplus E^s(x)$, which varies measurably with respect to $x$, is $F_{T,A}$-invariant, and such that $E^u$ is uniformly expanded and $E^s$ is uniformly contracted.

Uniform hyperbolicity of $(T,A)$ is equivalent to the following: there exists $c>0$, $\lambda>1$ such that $\| A_T^n(x) \| > c \lambda^n$ for $\mu$-a.e. $x$ and $n \ge 0$. See [@Yo proposition 2].

In this paper we address the question of the dependence of ${\mathit{LE}}(A,T)$ on the dynamics $T$, where $A\colon  X \to {\mathbb{G}}$ is fixed. We shall consider the following two general situations:

[*Measurable situation:*]{}

:   Assume that $(X,\mu)$ is a non-atomic Lebesgue space and $A \colon X \to {\mathbb{G}}$ is a bounded measurable map. The dynamics $T$ varies in the space ${\mathit{Aut}}(X,\mu)$ of the automorphisms of $(X,\mu)$ (i.e., bi-measurable $\mu$-preserving bijections). We will always consider the space ${\mathit{Aut}}(X,\mu)$ endowed with the *weak topology*, according to which $T_n \to T$ iff $\mu(T_n(B) \mathbin{\vartriangle} T(B)) \to 0$ for every measurable set $B \subset X$.

[*Continuous situation:*]{}

:   $X$ is a compact manifold of dimension at least $2$, $\mu$ is a volume measure, and is continuous. Now the dynamics $T$ varies in the space ${\mathit{Homeo}}(X,\mu)$ of $\mu$-preserving homeomorphisms, which we endow 