# Refined equidistribution for the periodic trajectories of Anosov flows?

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 $t \to \infty$.

• For the general dynamical situation, Bowen proved the equidistribution result for grouping over lengths $\leq T$; see ams.org/mathscinet-getitem?mr=298700 -- I initially wrote this as an answer and then realized that it's not an answer to your question because you are asking about grouping together orbits of a fixed length. Oct 12, 2016 at 21:13
• Do you mean equidistribution with respect to the Bowen-Margulis measure?
– Asaf
Oct 12, 2016 at 21:18
• @Asaf: Yes, precisely as in the general equidistribution theorems. Perhaps, as the spectrum of lengths is simple in the generic situation (in which case the question reduces to equidistribution of long geodesics), it should be possible to construct counterexamples? Oct 12, 2016 at 21:47
• ...but then we could modify the problem just by groupping together the geodesics of lengths belonging to the interval $[L,L+1]$. Oct 12, 2016 at 21:59
• For what it's worth, the [L,L+1] question is answered in Bowen's paper; I undeleted my original answer and edited it to indicate this. As for your original question on specifying the length exactly, I suspect you may be right that one can construct counterexamples, but I don't know for sure one way or the other. Oct 13, 2016 at 0:02

Since you mention the related question of grouping together geodesics of lengths in the interval $[L,L+1]$, let me point out that this case is covered by the 1972 result of Bowen that I mentioned in the comments: [R. Bowen, "Periodic orbits for hyperbolic flows", Amer. J. Math 94 (1972), 1-30].
Bowen proved that if $M$ is a compact Riemannian manifold and $\psi_t$ is an Axiom A flow, then on each basic set, the periodic orbits (of length $\leq T$) are equidistributed with respect to the unique measure of maximal entropy on that basic set. In particular, this holds for any transitive Anosov flow (in which case the basic set is the whole manifold), which includes geodesic flow in negative curvature.
In fact, (5.5) from that paper shows that if the flow is "C-dense" (basically this just guarantees that it's not a constant time suspension, and in particular C-density holds for geodesic flow in negative curvature), then for any $\epsilon>0$, if one considers the measures $\omega_{\epsilon,t}$ obtained by averaging closed geodesics with length in $[t-\epsilon,t+\epsilon]$, then one has $\omega_{\epsilon,t}\to \mu$, the unique MME, as $t\to\infty$.
• Thank you! I wasn't aware that Bowen's result was that precise. I'm afraid, though, that I had overlooked the revised question as well, and should rather be looking at intervals such as $[\log{L}, \log{(L+1)}]$ rather than $[L,L+1]$ - which would be much, much more precise. Hence I am interested in how much one can push Bowen's method by replacing the $\epsilon$ by functions $c(t)$ going to zero. Oct 13, 2016 at 0:52
• (I had neglected that, the growth rate being exponential for the number of geodesics of length $\leq T$, Duke's theorem would correspond with something as strong as $c(t) = \exp(-t)$, whereas having the result in each segment $[L,L+1]$ only corresponds with an averaged result about quadratic orders with discriminant in a dyadic interval.) Oct 13, 2016 at 1:16
Your question should follow from an equidistribution result of periods less than $$T$$ with an exponential error term like $$e^{-\epsilon T}$$. Then we can deduce the equisditribution result of periods in $$[T,T+c(T)]$$ with $$c(T)$$ decays no faster than $$e^{-\epsilon T/2}$$. This kind of exponential equidistribution results is known in homogeneous cases, see for example the references in my paper https://arxiv.org/abs/2202.08323. For general Anosov flows, the counting with exponential error term is mainly due to Dolgopyat. For equidistribution, maybe you can find it in some works of Pollicott.