  In the recent work arXiv:1311.3999, the authors proved that real analytic
manifolds $(M, g)$ with maximal eigenfunction growth must have a self-focal
point p whose first return map has an invariant L1 measure on $S^*_p M$. In
this addendum we add a purely dynamical argument on circle maps to improve the
conclusion to: all geodesics from p are smoothly closed.
