  A Boolean function $f:V \to \{-1,1\}$ on the vertex set of a graph $G=(V,E)$
is locally $p$-stable if for every vertex $v$ the proportion of neighbours $w$
of $v$ with $f(v)=f(w)$ is exactly $p$. This notion was introduced by Gross and
Grupel in [1] while studying the scenery reconstruction problem. They give an
exponential type lower bound for the number of isomorphism classes of locally
$p$-stable functions when $G=Q_n$ is the $n$-dimensional Boolean hypercube and
ask for more precise estimates. In this paper we provide such estimates by
improving the lower bound to a double exponential type lower bound and finding
a matching upper bound. We also show that for a fixed $k$ and increasing $n$,
the number of isomorphism classes of locally $(1-k/n)$-stable functions on
$Q_n$ is eventually constant. The proofs use the Fourier decomposition of
functions on the Boolean hypercube.
