  Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor
sets and by similar results of Knutsen for joint time-frequency representations
(i.e., the short-time Fourier transform (STFT) with a Gaussian window,
equivalent to Fock spaces), we suggest a general setting relating localization
and uncertainty and prove, within this context, an uncertainty principle for
Cantor sets in Bergman spaces on the unit disk, where the Cantor set is defined
as a union of annuli that are equidistributed in the hyperbolic measure.The
result can be written in terms of analytic Cauchy wavelets. As in the case of
the STFT considered by Knutsen, our result consists of a two-sided bound for
the norm of a localization operator involving the fractal dimension log 2 / log
3 in the exponent. As in the STFT case and in Dyatlov fractal uncertainty
principle, the (hyperbolic) measure of the dilated iterates of the Cantor set
in the disk tends to infinity, while the corresponding norm of the localization
operator tends to zero.
