Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension

Question

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?

Answer 1

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 $[3-\epsilon,3]$).

Answer 2

1. Show that for a Bernoulli system, there exists ergodic (Bernoulli) measures of any given entropy (between 0 and full entropy). Pick such a measure with appropriate entropy as you would like. Recall that especially in such systems, the entropy relates to the Minkowski dimension (and also the Hausdorff dimension, by a result of Furstenberg).
2. Use the ergodic theorem a-la Furstenberg to generate a generic point whose orbit is dense in the support.
3. Use a Bernouliocity theorem (a-la Adler-Weiss) to transfer everything to the modular surface (such theorem is achieved in practice by constructing actual Markov partitions, so you save the metric structure by preserving entropy).

There is a minor technicality in the fact that you might need to use countable rather than finite encoding (due to the fact your lattice might be non-uniform), so you would need to massage a bit of the arguments, but everything is essentially well-known (and actually, this encoding method show you you are able to choose geodesics with bounded height so by Dani's correspondence, you may only deal with endpoints which are BA). Some good sources for the encoding (for the modular surface) are C. Series' articles, or S. Katok's book. There is a chance that Series' articles are actually dealing with the general case.