Summary: 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.