# Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension

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

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 $$[3-\epsilon,3]$$).
• 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