Duke, and Linnik before him under a restrictive condition, proved that the set of closed geodesics of a given length $L$ is equidistributed on the modular surface as $L \to \infty$. This is a theorem of a deep arithmetic significance, which is powered by Siegel's ineffective theorem (but is semi-effective in the sense that it gives an effective equidistribution if one restricts to a sequence of orders in real quadratic fields $\mathbb{Q}(\sqrt{D})$ fulfilling $L(1,\chi_D) > D^{-0.1}$, say.)

The geodesic flow on the modular surface is a particular case of geodesic flow in Riemannian manifolds of negative curvature, or more generally, of Anosov (or Axiom A) flows. In such a situation, starting with Margulis's thesis and, subsequently, taking inspiration from Selberg's trace formula and from the distribution of prime ideals in number fields, various growth and equidistribution theorems have been obtained for the periodic trajectories of a bounded length. For the case of the modular surface, however, these results amount to averaged formulas over quadratic fields of a bounded discriminant, which are much less deep. The asymptotics of the number of closed geodesics of length $\leq T$ on the modular surface was proved by Siegel in 1944 and was, or could have been, known already to Gauss, in a different language of quadratic forms.

**Question.** *For Anosov flows - say, for concreteness, just the case of the closed geodesics on a complete negatively curved surface of finite volume, - and for $c(t) =\exp(-t)$ or at least for $c(t) = 1/t$, is the set of finite trajectories of period belonging to $[T,T+c(T)]$ expected to be equidistributed in the Bowen-Margulis measure as $T \to \infty$*?

Here, one definitely has to do some grouping by period; already with Linnik's problem on the modular surface, it is not true that individual closed geodesics need to equidistribute as the length approaches infinity. The Bowen-Margulis theorem gives the answer 'in the average', as one further groups the geodesics over all lengths $\leq T$. Vaughn Climenhaga's answer shows that $c(t) = \epsilon$ is admissible, for all $\epsilon > 0$, and the question is how much may one expect to refine this, in this generality, to functions $c(t) \to 0$.

In the case of the modular surface, Bowen's $c(t) = \epsilon$ result only treats real quadratic orders of discriminants belonging to a dyadic segment $[e^T,Ce^T]$, for arbitrary $C > 1$ and $T \gg_C 1$, whereas Duke's much more precise theorem would correspond with an $c(t)$ as small as exponential in $-t$. I would be interested in any work that considers a refinement of the equidistribution result to a function $c(t) \to 0$ as $t \to \infty$.

fixedlength. $\endgroup$ – Vaughn Climenhaga Oct 12 '16 at 21:13