  Let $\mathcal H$ be a finite dimensional complex Hilbert space with dimension
$n \ge 3$ and $\mathcal P(\mathcal H)$ the set of projections on $\mathcal H$.
Let $\varphi: \mathcal P(\mathcal H) \to \mathcal P(\mathcal H)$ be a
surjective map. We show that $\varphi$ shrinks the joint spectrum of any two
projections if and only if it is joint spectrum preserving for any two
projections and thus is induced by a ring automorphism on $\mathbb C$ in a
particular way. In addition, for an arbitrary $k \ge 3$, $\varphi$ shrinks the
joint spectrum of any $k$ projections if and only if it is induced by a unitary
or an anti-unitary. Assume that $\phi$ is a surjective map on the Grassmann
space of rank one projections. We show that $\phi$ is joint spectrum preserving
for any $n$ rank one projections if and only if it can be extended to a
surjective map on $\mathcal P(\mathcal{H})$ which is spectrum preserving for
any two projections. Moreover, for any $k >n$, $\phi$ is joint spectrum
shrinking for any $k$ rank one projections if and only if it is induced by a
unitary or an anti-unitary.
