  We study the transverse dynamics of two-dimensional traveling periodic waves
for the gravity--capillary water-wave problem. The governing equations are the
Euler equations for the irrotational flow of an inviscid fluid layer with free
surface under the forces of gravity and surface tension. We focus on two open
sets of dimensionless parameters $(\alpha,\beta)$, where $\alpha$ and $\beta$
are the inverse square of the Froude number and the Weber number, respectively.
For each arbitrary but fixed pair $(\alpha,\beta)$ in one of these sets,
two-dimensional traveling periodic waves bifurcate from the trivial constant
flow. In one open set we find a one-parameter family of periodic waves, whereas
in the other open set we find two geometrically distinct one-parameter families
of periodic waves. Starting from a transverse spatial dynamics formulation of
the governing equations, we investigate the transverse linear instability of
these periodic waves and the induced dimension-breaking bifurcation. The two
results share a common analysis of the purely imaginary spectrum of the
linearization at a periodic wave. We apply a simple general criterion for the
transverse linear instability problem and a Lyapunov center theorem for the
dimension-breaking bifurcation. For parameters $(\alpha,\beta)$ in the open set
where there is only one family of periodic waves, we prove that these waves are
linearly transversely unstable. For the other open set, we show that the waves
with larger wavenumber are transversely linearly unstable. We also identify an
open subset of parameters for which both families of periodic waves are
tranversely linearly unstable. For each of these transversely linearly unstable
periodic waves, a dimension-breaking bifurcation occurs in which
three-dimensional doubly periodic waves bifurcate from the two-dimensional
periodic wave.
