  For an infinitely renormalizable negative Schwarzian unimodal map $f$ with
non-flat critical point, we analyze statistical properties of periodic points
as the periods tend to infinity. Introducing a weight function $\varphi$ which
is a continuous or a geometric potential $\varphi=-\beta\log|f'|$
($\beta\in\mathbb R$), we establish the level-2 Large Deviation Principle for
weighted periodic points. From this, we deduce that all weighted periodic
points equidistribute with respect to equilibrium states for the potential
$\varphi$. In particular, it follows that all periodic points are
equidistributed with respect to measures of maximal entropy, and all periodic
points weighted with their Lyapunov exponents are equidistributed with respect
to the post-critical measure supported on the attracting Cantor set.
