A famous theorem of Tate implies that two smooth projective curves over a finite field have the same zeta function if and only if their Jacobians are isogenous (in particular, the curves needn't be isomorphic). We prove that two smooth projective curves are isomorphic (up to automorphisms of the ground field) if and only if certain associated dynamical systems (arising from class field theory) are topologically conjugate. This is in turn equivalent to an equality of all Dirichlet L-series of the curves via a group isomorphism between the groups of linear characters of their absolute Galois groups.