9 problems

Let $X$ be the quotient under consideration, let $h(t)$ denote the horocycle flow, let $p\in X$ be non-periodic, and let $\mu_X$ be the invariant measure. For a sequence $a_n$, the…

Let a horocycle flow be given on a finite-measure dynamical system, and let a sufficiently smooth time change be a smooth reparametrization of that flow. A unitary Koopman represen…

Let $\Gamma\subset PSL(2,\mathbb{R})$ be such that $X=\Gamma\backslash PSL(2,\mathbb{R})$ is not compact, and let $(h_t)$ be the horocycle flow acting on $X$. Margulis–Shah conject…

Let $G=PSL_2(\mathbb{R})$, let $\Gamma$ be a lattice in $G$, and set $X=\Gamma\backslash G$. Let $\mu_X$ be the normalized projection of Haar measure to $X$, and let…

Let $\mathcal{O}_v$ denote the distributions appearing in the expansion of the horocycle integral $\gamma_x(\varphi,T)$, and let the horocycle flow act on distributions through its…

Let $M$ carry an Anosov flow $g_\tau$ and a horocycle flow $h_\rho$, and let $\tau(\rho,\alpha,x)$ be the renormalization time defined by … For a function $\varphi$ and $T>0$, writ…

Horocycle-flow obstruction conjecture. The natural analogues of those theorems should hold for the horocycle flow, with an infinite countable family of obstructions.

Let $S$ be a hyperbolic surface of infinite volume, represented as $S=\Gamma\backslash\mathbb{H}$, and let the horocycle flow act on its unit tangent bundle. Let…

A horocycle flow on a compact negatively curved surface has countable Lebesgue spectrum, and consider a time-change satisfying Kuschnirenko's condition. Katok's conjecture. Countab…