Dynamical L-functtons and Homology of Closed Orbits

Dirichlet L-function density theorem for primes in arithmetic progression i i Dynamical L-function density theorem for closed orbits in homology class In the dynamical case, however, the "ideal class group" (= the first integral homology group) might have infinite order, so that some extra phenomena will be seen. To fix our terminology, we let {,} be a smooth, transitive Anosov flow [4] on a closed manifold X. We assume that 4>t has the weak-mixing property [17]. We denote by h the topological entropy of t, and by ju a measure of maximal entropy on X, that is, an invariant probability measure whose metric entropy h^ equals h. It is known that there exists exactly one measure with ftp = h [20]. The canonical winding cycle O, which measures the average of "homological" direction in which the orbits of the flow are traveling, is defined by