TITLE: Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension
QUESTION [11 upvotes]: 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?

REPLY [2 votes]: 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).
Use the ergodic theorem a-la Furstenberg to generate a generic point whose orbit is dense in the support.
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.