  Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain,
that is, a domain which satisfies interior Corkscrew and Harnack Chain
conditions (these are respectively scale-invariant/quantitative versions of the
openness and path-connectedness), and whose boundary $\partial\Omega$ is
$n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric
divergence form elliptic operators and let $\omega_{L_0}$, $\omega_L$ be the
associated elliptic measures. We show that if $\omega_{L_0}\in
A_\infty(\sigma)$, where $\sigma=H^n|_{\partial\Omega}$, and $L$ is a
perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$
satisfies certain Carleson measure condition), then $\omega_L\in
A_\infty(\sigma)$. Moreover, if $L$ is a sufficiently small perturbation of
$L_0$, then one can preserve the reverse H\"older classes, that is, if for some
$1<p<\infty$, one has $\omega_{L_0}\in RH_p(\sigma)$ then $\omega_{L}\in
RH_p(\sigma)$. Equivalently, if the Dirichlet problem with data in
$L^{p'}(\sigma)$ is solvable for $L_0$ then so it is for $L$. These results can
be seen as extensions of the perturbation theorems obtained by Dahlberg,
Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a
consequence of our methods we can show that for any perturbation of the
Laplacian (or, more in general, of any elliptic symmetric operator with
Lipschitz coefficients satisfying certain Carleson condition) if its elliptic
measure belongs to $A_\infty(\sigma)$ then necessarily $\Omega$ is in fact an
NTA domain (and hence chord-arc) and therefore its boundary is uniformly
rectifiable.
