  We prove the existence of local stable, unstable, and center manifolds for
stochastic semiflows induced by rough differential equations driven by rough
paths valued stochastic processes around random fixed points of the equation.
Examples include stochastic differential equations driven by a fractional
Brownian motion with Hurst parameter $H > \frac{1}{4}$. In case the top
Lyapunov exponent is negative, we derive almost sure exponential stability of
the solution.
