  For integer $m\ge3$, we study the dynamical system $(\Lambda_m,\sigma_m)$
where $\Lambda_m$ is the set $\{w\in\{0,1\}^\mathbb{N}: w$ does not contain
$0^m$ or $1^m\}$ and $\sigma_m$ is the shift map on $\{0,1\}^\mathbb{N}$
restricted to $\Lambda_m$, study the Bernoulli-type measures on $\Lambda_m$ and
find out the unique equivalent $\sigma_m$-invariant ergodic probability
measure. As an application, we obtain the Hausdorff dimension of the set of
univoque sequences, the Hausdorff dimension of the set of sequences in which
the lengths of consecutive $0$'s and consecutive $1$'s are bounded, and the
Hausdorff dimension of their frequency subsets.
