Invariant manifolds and stability for rough differential equations

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.