Stability of fixed points of Dirac structures

Given an $L_{\infty}$-algebra $V$ and an $L_{\infty}$-subalgebra $W$, we give sufficient conditions for all small Maurer-Cartan elements of $V$ to be equivalent to Maurer-Cartan elements lying in $W$. As an application, we obtain a stability criterion for fixed points of a Dirac structure (for instance a twisted Poisson structure), i.e. points where the corresponding leaf is zero-dimensional. The criterion guarantees that any nearby Dirac structure also has a fixed point.