On the closure $\overline T$ of Teichmüller space, consider a continuous parametrization $\overline T \times \Sigma_A^+ \to \mathbb S^2$ such that Image ($t, \Sigma_A^+ = \Lambda(\Gamma_t)$). Is the Hausdorff dimension of $\Lambda(\Gamma_t)$ continuous in $t \in \overline T$?
(Note, this problem was added by the editor to the end of Rufus' last paper [1])
For a Kleinian group $\Gamma$, is the Hausdorff dimension of $\Lambda (\Gamma) <2$ if $\Lambda(\Gamma)$ is not the whole sphere?
(Note, this problem was added by the editor to the end of Rufus' last paper [1])
$\varphi _t: T^1M \to T^1M$ Anosov geodesic flow and $V: M \to \mathbb{R}$ such that $\int V(\pi \varphi_t x ) = 0 $ on every closed geodesic. Is $V$ identically $0$?
$\Gamma$ a Kleinian group with limit set $\Lambda$. Specification when there are no parabolic nor elliptic elements.
(Guillemin-Kazhdan) Invariant distributions for geodesic flows. Are they approximated by periodic orbit measures?
Horocycle and geodesic flows for $SL(2,R)/SL(2,Z)$:
-- min u.e. almost
-- something about symbolic dynamics and continued fractions?
See the entropy of the geodesic flow as the rate of growth of eigenfunctions for some operator in momentum space. Related to Fourier transform of Laplacian on the manifold?
Let $\mathcal M$ be the space of Riemannian metrics on $M$ with volume 1. What is $\{ h(\varphi_1), \varphi {\textrm { geodesic flow of }}\; g: g \in \mathcal M\} $? What is the relation with topological invariants of $M$?
If a geodesic flow is expansive, is it an Anosov flow?
Fixing compact manifold $M$, what are the possible behaviors of the geodesic flows for all Riemannian metrics? For instance, if $\pi _1(M) = 0 ,$ does some geodesic flow have entropy 0?
Interpret $-\log \lambda^u$ as a potential function (Kolmogorov's idea on surfaces of negative curvature)?
Nonalgebraic Anosov diffeos. Classify 3-dimensional Anosov flows; Is the variable curvature surface geodesic flow conjugate to constant curvature?
