Let p and q be distinct primes. In a joint work with Payman Kassaei (King’s College) we are studying questions concerning the dynamics of the Hecke operator T_q in its action on the modular curves X_1(N), where (N, pq) = 1, in the p-adic topology. Using Serre-Tate coordinates and isogeny volcanoes for points of X_1(N) with ordinary reduction, and using the graphs of supersingular elliptic curves and the Gross-Hopkins period map for points with supersingular reduction, we are able to get rather precise understanding of orbits of points. Our work and the work of Herrero, Menares and Rivera-Letelier seem to complement each other.