Problem 134
If $\varphi _t$ is flow on a homogeneous space $G/ \Gamma$ with positive entropy, then there exists a compact $\varphi _t $ invariant section for the action of $N$ If the flow has entropy 0 and is ergodic, does this mean that there is no $N$?
Comments
(For part 1.) Hardly legible. Wild guess here. Don’t see what that means.
(part 2) Is $\varphi _t $ a one parameter subgroup here? and $N$ associated to it?
In the setting where $G = SL(2, \mathbb R)$, Yitwah Cheung and I built a section to the horocycle flow for $\Gamma = SL(2, \mathbb Z)$, and with Jon Chaika and Samuel Lelievre built one for $\Gamma = \Delta(2, 5, \infty)$. This construction was generalized by Uyanik-Work, and Sarig-Schmoll showed how to realize the horocycle flow as a suspension flow over an adic transformation.