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?

  • $\begingroup$ What do you call a geodesic on $X$? An orbit of the geodesic flow? $\endgroup$ – YCor Oct 8 '18 at 8:02
  • $\begingroup$ @YCor Yes, that's right $\endgroup$ – Kim Oct 8 '18 at 8:06
  • $\begingroup$ 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? $\endgroup$ – Selim G Oct 17 '18 at 15:56
  • $\begingroup$ @SelimG, this indeed the case, which you can show by Bernoullocity theorem (which here can be done directly). There are minor differences (invertability, which you can overcome by moving to a solonoioid) and the fact that the geodesic flow is Anosov rather than expansive map, but technically this is the same. It is more analogous to the case of hyperbolic toral automorphism. $\endgroup$ – Asaf Oct 18 '18 at 23:18

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

  • $\begingroup$ 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? $\endgroup$ – Kim Oct 16 '18 at 20:50
  • $\begingroup$ @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. $\endgroup$ – Ian Agol Oct 16 '18 at 20:58
  • $\begingroup$ 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? $\endgroup$ – Kim Oct 17 '18 at 5:53
  • $\begingroup$ Maybe the geodesics that lie outside of some cusp? Maybe one can do a similar thing on a closed surface. $\endgroup$ – Ian Agol Oct 17 '18 at 11:07
  • $\begingroup$ @Kim: I suspect using Theorem 1.4 of this paper, one ought to be able to show that the set of geodesics outside a fixed cusp in the modular surface has Hausdorff dimension between 1 and 3. Presumably if this dimension varies continuously, one can achieve all dimensions in between. mathscinet.ams.org/mathscinet-getitem?mr=3299603 $\endgroup$ – Ian Agol Oct 17 '18 at 23:08
  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.

  • $\begingroup$ Could you give some references for these steps? For example, the connection between entropy and Hausdorff dimension, and the other papers you mention. $\endgroup$ – Kim Oct 19 '18 at 22:26
  • $\begingroup$ The connection between entropy and dimension is immediate (say by the definition of topological entropy, and there's some tricks (i.e. the variational principle) showing you can realize that by suitable measure). The easiest example is the half-half Bernoulli measure on the Cantor set, leading to entropy of ln2, while the dimension is ln2/ln3 (and the alphabet is 3 letters). This would be the case for example for any Bernoulli system and then you can generate appropriate orbits easily via the symbolic model. [Actually, in any positive entropy system by lifting from the Bernoulli factor]. $\endgroup$ – Asaf Oct 21 '18 at 16:06
  • $\begingroup$ About the papers for the coding, see C. Series - 'Symbolic dynamics for geodesic flows', you can also try the introduction here - londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms/… , Katok's book about continued fractions is named 'Fuchsian Groups'. The Adler-Weiss argument (in the case of geodesic flows) is carried in Ornstein-Weiss - link.springer.com/article/10.1007/BF02762673 , but Benjy told me a few years ago that it is actually due to Adler. $\endgroup$ – Asaf Oct 21 '18 at 16:11

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