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.

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