Periodic Orbits and Holonomy for Hyperbolic Flows

[14]. More precisely, we have this error term if we can choose three closed orbits of least periods l1, l2, l3 such that θ = (l1 − l2)/(l2 − l3) is diophantine, i.e., there exists C > 0 and β > 0 such that |qθ − p| ≥ Cq−(1+β) for all p ∈ Z and q ∈ N. In this paper we shall consider compact groups extensions of hyperbolic flows. Let G be a compact Lie group. Let φ̂t : Λ̂ → Λ̂ be a topologically weak mixing G-extension of a hyperbolic flow φt : Λ → Λ with projection π : Λ̂ → Λ. Given a closed φ-orbit and x ∈ τ there exists a unique element g ∈ G such that, for x̂ ∈ π−1(x), φl(τ)x̂ = gx̃. If we choose another point x′ ∈ τ then the corresponding group element is conjugate to g. We call the conjugacy class [g] in G the holonomy class of τ , which we denote by [τ ]. In [11], the following general equidistribution result was established: if χ ∈ Ĝ is a non-trivial character then