I am attending Thi’s talk at the Shanghai Institute of Mathematics and Interdisciplinary Sciences in Shanghai. Thi says this topic is her speciality.
Consider a compact hyperbolic manifolds given as \(\Gamma \backslash \mathbb{H}^{2}\) where \(\Gamma \subseteq \mathrm{PSL}(2,\mathbb{R})\) is cocompact. We consider the geodesic flow this surface. This is given by the action of \[\begin{equation} \Gamma \backslash \mathrm{PSL}(2,\mathbb{R}) \curvearrowleft \begin{bmatrix} e^{t} & \\ & e^{-t} \\ \end{bmatrix} \text{ for t } \in \mathbb{R} \end{equation}\]
Now suppose \(\Gamma\) is non-compact. What is the geodesic flow in the case of pair of pants?
Here are some definitions.
Say a flow \(\Phi^{t}: \Omega \curvearrowleft\) is topologically mixing if \(\forall U,V \subseteq \Omega\) open, non-empty one has some \(T>0\) such that for any \(t > T\) we get \[\begin{equation} \Phi^{t}( U) \cap V \neq \emptyset \end{equation}\]
(DAL’BO 2000)
Let \(\Gamma\) be non-elementary (non-trivial, not generated by 1 element). Then \(g^{t}\) (geodesic flow) is mixing on its non-wandering set if and only if \(\oplus_{\gamma \in \Gamma} \mathbb{Z} l(\gamma) = \mathbb{R}\), where \(l(\gamma)\) is the length of a closed geodesic \(\gamma \in \mathrm{PSL}(2,\mathbb{R})\)
This theorem is important because it easily allows us to check this topological mixing property.:w
For a geodesic flow, the non-wandering set is just the union of periodic geodesics.
We take \(G = \mathrm{SL}(d,\mathbb{R})\) and \(K = \mathrm{SO}(d,\mathbb{R})\). We put \(A \subseteq \mathrm{SL}(d,\mathbb{R})\) to be the diagonal matrices with positive entries. Lastly \(M = Z_{k}(A)\) is the sign group i.e. \(\mathrm{diag}( \pm 1, \pm 1,\dots , \pm 1)\).
Let \(\Gamma \subseteq G\) be a Zariski dense subgroup.
We set \(\mathfrak{a} = \log A\) to be the Lie algebra. This is the set of traceless diagonal matrices. We denote \(\mathfrak{a}^{+} \subseteq \mathfrak{a}\) to be the set of \(\mathrm{diag}(t_{1},t_{2},\dots,t_{d})\) such that \(t_{1} \geq \dots \geq t_{d}\). We denote \(\mathfrak{a}^{++} \subseteq \mathfrak{a}^{+}\) to be the subset when the inequalities are strict. This last set is called the positive Weyl chamber.
For \(y \in \mathfrak{a}^{+}\), \(y \neq 0\), the action \(\Gamma \backslash G / M \curvearrowleft e^{tY}\) is called a Weyl chamber flow.
We say it’s regular when \(Y \in \mathfrak{a}^{++}\).
When \(\Gamma\) is a lattice, use Howe-Moore (1979). This implies that the Weyl chamber flow is topologically mixing for all \(y \in \mathfrak{a}^{+} \setminus \{0\}\).
We want higher rank length. We define the Jordan projection as the map \[\begin{align} \lambda : G & \rightarrow \mathfrak{a}^+ \\ g & \mapsto ( \log |\Lambda_{1}(g)|, \dots, \log |\Lambda_{d}(g)|) \end{align}\] These \(\{\Lambda_{i}(g)\}\) are complex eigenvalues of \(G\).
Then, when \(\Gamma \subseteq G\) is Zariski dense, then there is
(Kim, 2006)
(Benoist, 2000)
The following is siffucient for topologically mixing. \[\begin{equation} \oplus_{\gamma \in \Gamma} \mathbb{Z} \gamma(\Gamma) = \mathfrak{a} \end{equation}\]
Let us now consider \[\begin{equation} \mathcal{C}(\Gamma) = \cup_{ \gamma \in \Gamma} \mathbb{R}_{>0} \lambda(\gamma). \end{equation}\] We have the following theorem.
(97, Benoist)
The set \(\mathcal{C}(\Gamma)\) is closed, convex and has non-empty interior.
The following was Thi’s doctoral work (in collaboration with Olivier).
(D.-Glorieux 2020)
Let \(\Gamma \subseteq G\) be Zariski dense. Let \(y \in \mathfrak{a}^{++} \setminus {0}\). Then \(e^{t Y}\) is mixing on its ``non-wanishing’’ set if and only if \(y \in \mathcal{C}^{0}(\Gamma)\).
When \(y \in \mathfrak{a}^{++}\), \(y \notin \mathcal{C}(\Gamma)\) all trajectories diverge.
There are new results by Oh, Edwards and Lee which are more effective.
Our proof is fairly elementary. We have \[\begin{equation} \Omega = \cup_{\substack{\gamma \in \Gamma \\ \text{ loxodromic }}} \Gamma h_{\gamma} A M \end{equation}\]
Some \(g \in G\) is loxodromic when \(\lambda(g) \in \mathfrak{a}^{++}\). And then \[\begin{equation} g = h_{h} e^{\lambda(g)} m_{g} h_{g}^{-1}, \end{equation}\] where \(m_{g} \in M\) and \(h_{g} \in G\).
(Conze-Guivarc’h 2002)
For \(\Omega\) closed, \(A\)-invariant, \(\exists\) a dense orbit for \(\Omega \curvearrowleft A\).
If we have topological mixing, we have in particular \[\begin{equation} \Gamma g A M \subseteq \overline{\{ \Gamma h e^{t y} M \}_{t \in \mathbb{R}} }. \end{equation}\] Then \(\exists (\gamma_{m}) \subseteq \Gamma\) such that \[\begin{equation} h e^{v + t_{n} y} m h^{-1} \simeq \gamma_{n} \end{equation}\] And then \(\lambda (\gamma_{m}) \simeq v+ t_{n} y\). This is related to the ping-pong principle somehow (I dont’ understand).
This page was updated on April 30, 2026.
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