  We introduce a parametrised family of maps $\{S_{\eta}\}_{\eta \in [1,2]}$,
called symmetric doubling maps, defined on $[-1,1]$ by $S_\eta (x)=2x-d\eta$,
where $d\in \{-1,0,1 \}$. Each map $S_\eta$ generates binary expansions with
digits $-1$, 0 and 1. We study the frequency of the digit 0 in typical
expansions as a function of the parameter $\eta$. The transformations $S_\eta$
have a natural ergodic invariant measure $\mu_\eta$ that is absolutely
continuous with respect to Lebesgue measure. The frequency of the digit 0 is
related to the measure $\mu_{\eta}([-\frac12,\frac12])$ by the Ergodic Theorem.
We show that the density of $\mu_\eta$ is piecewise smooth except for a set of
parameters of zero Lebesgue measure and full Hausdorff dimension and give a
full description of the structure of the maximal parameter intervals on which
the density is piecewise smooth. We give an explicit formula for the frequency
of the digit 0 in typical signed binary expansions on each of these parameter
intervals and show that this frequency depends continuously on the parameter
$\eta$. Moreover, it takes the value $\frac23$ only on the interval $\big[
\frac65, \frac32\big]$ and it is strictly less than $\frac23$ on the remainder
of the parameter space.
