  In this paper, we show the following: the Hausdorff dimension of the spectrum
of period-doubling Hamiltonian is bigger than $\log \alpha/\log 4$, where
$\alpha$ is the Golden number; there exists a dense uncountable subset of the
spectrum such that for each energy in this set, the related trace orbit is
unbounded, which is in contrast with a recent result of Carvalho (Nonlinearity
33, 2020); we give a complete characterization for the structure of gaps and
the gap labelling of the spectrum. All of these results are consequences of an
intrinsic coding of the spectrum we construct in this paper.
