Consider the geodesic flow on $X = \Gamma \backslash \text{PSL}(2,\mathbf{R})$, the unit tangent bundle of a hyperbolic surface, where $\Gamma$ is a lattice.

I have heard that, for any real number $\alpha \in [1,3]$, there exists a orbit of the geodesic flow whose closure in $X$ has Hausdorff dimension $\alpha$. Where can I find a proof of this?

This question has an open bounty worth +50 reputation from Kim ending tomorrow.

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We request a complete detailed proof, or a reference which gives such a proof.

  • What do you call a geodesic on $X$? An orbit of the geodesic flow? – YCor Oct 8 at 8:02
  • @YCor Yes, that's right – Kim Oct 8 at 8:06
  • I think a related question, which encapsulate all the difficulties of yours but lowers the dimension, is the following: can one find orbit closures of the doubling map $x \in [0,1[ \mapsto 2x \ mod \ 1$ of arbitrary (between 0 and 1) Hausdorff dimension? – Selim G 3 hours ago

I did a bit of a literature search and couldn't find much.

Let $\Sigma$ be a closed surface, and $PT\Sigma$ the projective tangent bundle (double covered by the unit tangent bundle $X$). Lenzhen and Souto show in Theorem 1.3 that there exists a closed subset $\mathbb{X}(\Sigma)\subset PT\Sigma$ invariant under the geodesic flow such that $1< dim_{\mathcal{H}} \mathbb{X}(\Sigma)<3$ (at least for certain hyperbolic surfaces $\Sigma$ with small systole).

However, the set $\mathbb{X}(\Sigma)$ is not the closure of a geodesic in $\mathbb{X}(\Sigma)$. This follows from the description (2.1), where the geodesics in $\mathbb{X}(\Sigma)$ have ends which are asymptotic to laminations (see Figure 1). So the closure of each geodesic will just be itself together with the lamination it limits to.

In section 5 of the paper, they describe invariant subsets $X^\tau \subset PT\Sigma, \tau \in [0,\infty]$, and show that $dim_{\mathcal{H}} X^0=1, dim_{\mathcal{H}} X^\infty = 3$. They ask if the function $\tau \to dim_{\mathcal{H}} X^\tau$ is continuous? If the answer is yes, and if $X^\tau$ is the closure of a single geodesic for $\tau >0$, then it would imply a positive answer to your question (at least in an interval $[3-\epsilon,3]$).

  • These results are more recent than what I was expecting. I had assumed my question was about a piece of well-known folklore. Is this not the case? – Kim 22 hours ago
  • @Kim: I’m not exactly in the field, so if it is folklore, I’m not the folk that would know of it. It’s possible that a much more general result may be known for Anosov flows, but I didn’t find anything. If the authors knew of a result of this sort, it would seem reasonable that they would have included a reference. – Ian Agol 22 hours ago
  • The lack of a reference caught my attention as well. Do you know if anything like this is known for $\Gamma = \text{PSL}(2,\mathbf{Z})$ in particular? – Kim 13 hours ago
  • Maybe the geodesics that lie outside of some cusp? Maybe one can do a similar thing on a closed surface. – Ian Agol 8 hours ago

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