  We describe the idempotent Fourier multipliers that act contractively on
$H^p$ spaces of the $d$-dimensional torus $\mathbb{T}^d$ for $d\geq 1$ and
$1\leq p \leq \infty$. When $p$ is not an even integer, such multipliers are
just restrictions of contractive idempotent multipliers on $L^p$ spaces, which
in turn can be described by suitably combining results of Rudin and And\^{o}.
When $p=2(n+1)$, with $n$ a positive integer, contractivity depends in an
interesting geometric way on $n$, $d$, and the dimension of the set of
frequencies associated with the multiplier. Our results allow us to construct a
linear operator that is densely defined on $H^p(\mathbb{T}^\infty)$ for every
$1 \leq p \leq \infty$ and that extends to a bounded operator if and only if
$p=2,4,\ldots,2(n+1)$.
