to $T$ such that ${\mathit{LE}}(A,\tilde{T})<{\varepsilon}$.

In fact, it is easy to strengthen theorem \[t.cont\] by demanding only the existence of periodic pseudo-orbits with similar properties; see § \[ss.contdich\].

Let us call a periodic point $p$ of period $n$ for an homeomorphism $T \colon X\to X$ *persistent* if for every ${\varepsilon}>0$ there is $\delta>0$ such that if $\tilde{T}$ is an homeomorphism $\delta$-$C^0$-close to $T$ then $\tilde{T}$ has a periodic point $\tilde{p}$ of period $n$ which is ${\varepsilon}$-close to $p$. (For example, if $p$ is an isolated fixed point of Poincaré–Lefschetz index different from $1$ then $p$ is persistent.)

Theorem \[t.cont\], together with a semicontinuity property (proposition \[p.semiT\]), implies:

Let $T\in{\mathit{Homeo}}(X,\mu)$ and $A \in C^1(X, {\mathbb{G}})$. Assume that $T$ has a persistent periodic point $p = T^n(p)$ such that:

-   $A_T^n(p)$ is elliptic;

-   for some $i=0,1,\ldots,n-1$, the derivative $DA(T^i p)$ is non-zero.

Then there is a neighborhood ${\mathcal{U}}\subset {\mathit{Homeo}}(X,\mu)$ of $T$ and a residual subset ${\mathcal{R}}\subset {\mathcal{U}}$ such that ${\mathit{LE}}(A,\tilde{T}) = 0$ for all $\tilde T \in {\mathcal{R}}$.

Recall definition \[d.contdich\]. The $C^0$-generic map satisfies the continuous dichotomy – this follows easily from [@B] and proposition \[p.generic fubini\]. Here we extend this result to higher topologies:

\[t.contdich\] Let $1 \le r 