Geo/Horo

Problem 157

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])

References

1. [bowen1979hausdorff] Bowen R.  1979.  Publications Mathématiques de l'IHÉS. 50:11–25.

Problem 156

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])

References

1. [bowen1979hausdorff] Bowen R.  1979.  Publications Mathématiques de l'IHÉS. 50:11–25.

Problem 153

$\varphi _t: T^1M \to T^1M$ 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 $0$?

Problem 148

$\Gamma$ a Kleinian group with limit set $\Lambda$. Specification when there are no parabolic nor elliptic elements.

Problem 147

(Guillemin-Kazhdan) Invariant distributions for geodesic flows. Are they approximated by periodic orbit measures?

Problem 135

Horocycle and geodesic flows for $SL(2,R)/SL(2,Z)$:

-- min u.e. almost

-- something about symbolic dynamics and continued fractions?

Problem 129

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?

Problem 56

Is the horocycle flow an expansive flow?

Problem 38

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 $M$?

Problem 37

Are geodesic flows $h$-expansive?

Problem 36

If a geodesic flow is expansive, is it an Anosov flow?

Problem 30

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?

Problem 18

Interpret $-\log \lambda^u$ as a potential function  (Kolmogorov's   idea on surfaces of negative curvature)?

Problem 11

Nonalgebraic Anosov diffeos. Classify 3-dimensional Anosov flows; Is the variable curvature surface geodesic flow conjugate to constant curvature?