Focal points and sup-norms of eigenfunctions on analytic Riemannian manifolds II: the two-dimensional case

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 $S_p^{\ast }M$. In this addendum we add a purely dynamical argument on circle maps to improve the conclusion to: all geodesics from p are smoothly closed.