with the uniform ($C^0$) topology.

\[r.generic ergodicity\] In the respective topologies, the generic maps $T \in{\mathit{Aut}}(X,\mu)$ and $T \in {\mathit{Homeo}}(X,\mu)$ are *ergodic*. These are classical theorems of Halmos [@H] and Oxtoby and Ulam [@OU], respectively. Although we shall not use these results, we use some related ideas from [@AlPr].

More concretely, we are interested in finding out *under what conditions on the maps $A$ and $T$, is it possible to find a perturbation* (in one of the two topologies above, according to the situation considered) *$\tilde{T}$ of $T$ so that the Lyapunov exponent ${\mathit{LE}}(A,\tilde{T})$ drops to an arbitrarily small value or even to zero.* Moreover, we want those conditions to be *checkable*, instead of appealing to Baire’s theorem. It is clear that such conditions must exclude some kind of hyperbolicity: if for example $A$ is constant and hyperbolic, then ${\mathit{LE}}(A,T)$ is positive and independent of $T$. (We will see later that the “kinds” of hyperbolicity we have to exclude are not the same in the measurable and continuous situations.) With this in mind, we look for the weakest possible conditions that imply dichotomies between zero exponents and uniform hyperbolicity.

In the measurable situation, we shall define a condition over $A$, 