Anisotropic power-law k-inflation
  It is known that power-law k-inflation can be realized for the Lagrangian
$P=Xg(Y)$, where $X=-(\partial \phi)^2/2$ is the kinetic energy of a scalar
field $\phi$ and $g$ is an arbitrary function in terms of $Y=Xe^{\lambda
\phi/M_{pl}}$ ($\lambda$ is a constant and $M_{pl}$ is the reduced Planck
mass). In the presence of a vector field coupled to the inflaton with an
exponential coupling $f(\phi) \propto e^{\mu \phi/M_{pl}}$, we show that the
models with the Lagrangian $P=Xg(Y)$ generally give rise to anisotropic
inflationary solutions with $\Sigma/H=constant$, where $\Sigma$ is an
anisotropic shear and $H$ is an isotropic expansion rate. Provided these
anisotropic solutions exist in the regime where the ratio $\Sigma/H$ is much
smaller than 1, they are stable attractors irrespective of the forms of $g(Y)$.
We apply our results to concrete models of k-inflation such as the generalized
dilatonic ghost condensate/the DBI model and we numerically show that the
solutions with different initial conditions converge to the anisotropic
power-law inflationary attractors. Even in the de Sitter limit ($\lambda \to
0$) such solutions can exist, but in this case the null energy condition is
generally violated. The latter property is consistent with the Wald's cosmic
conjecture stating that the anisotropic hair does not survive on the de Sitter
background in the presence of matter respecting the dominant/strong energy
conditions.
