  We show that for the generic continuous maps of the interval and circle which
preserve the Lebesgue measure it holds for each k $\ge$ 1 that the set of
periodic points of period k is a Cantor set of Hausdorff dimension zero and of
upper box dimension one. Furthermore, building on this result, we show that
there is a dense collection of transitive Lebesgue measure preserving interval
map whose periodic points have full Lebesgue measure and whose periodic points
of period k have positive measure for each k $\ge$ 1. Finally, we show that the
generic continuous maps of the interval which preserve the Lebesgue measure
satisfy the shadowing and periodic shadowing property.
