We provide explicit bounds on the eigenvalues of transfer operators defined in terms of holomorphic data.

Abstract. For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to these of Dolgopyat [D2] for transitive Anosov flows on compact manifolds with C 1 jointly non-integrable horocycle foliations. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions, closed orbit counting functions, decay of correlations for Hölder continuous potentials. The situation considered here is substantially more difficult than the Anosov case since, even under the additional conditions, in general the geometry of the basic set can be rather complicated. 1

a priori bounds on transfer operator eigenvalues

Abstract. In this note we show that the length spectrum for metric graphs exhibit a very high degree of degeneracy. More precisely, we obtain an asymptotic for the number of pairs of closed geodesic (or closed cycles) with the same metric length. Let G = (V, E) be a finite graph with vertices V and edges E. We write E o for the set of oriented edges; for e ∈ E o, ē ∈ E o denotes the same unoriented edge with orientation reversed. (In the physics literature, the vertices are referred to as nodes and the edges as bonds.) The degree of a vertex is the number of outgoing