Orbits in the Maximal Ideal Space of H{infty}

Let H be the Banach algebra of bounded analytic functions inthe open unit disc D. We can define the rotation in the maximalideal space M(H). For a point x in M(H)\D, an orbit O(x) isnot closed in M(H). It is proved that there exists a point xin M(H) such that x is not contained in the Shilov boundaryX and cl O(x), the closure of O(x), contains X, and there existsa point y in M(H)\(D X) such that cl O(y) X. The rotationpresents many problems concerning H. The purpose of this paperis to discuss these problems.