We study axial (or odd-parity) perturbations about static and spherically
symmetric hairy black hole (BH) solutions in shift-symmetric DHOST (Degenerate
Higher-Order Scalar-Tensor) theories. We first extend to the family of DHOST
theories the first-order formulation that we recently developed for Horndeski
theories. Remarkably, we find that the dynamics of DHOST axial perturbations is
equivalent to that of axial perturbations in general relativity (GR) evolving
in a, distinct, effective metric. In the particular case of quadratic DHOST
theories, this effective metric is derived from the background BH metric via a
disformal transformation. We illustrate our general study with three examples
of BH solutions. In some so-called stealth solutions, the effective metric is
Schwarzschild with a shifted horizon. We also give an example of BH solution
for which the effective metric is associated with a naked singularity.
