论文标题
扁平投影束上的烟节流：拓扑备注和应用
Horocycle flow on flat projective bundles: topological remarks and applications
论文作者
论文摘要
在本文中，我们研究了双曲线表面上平坦的投影束上叶状烟节流动的动力学的拓扑方面，并引起了良好的后果。如果$ρ：γ\ to {\ rm psl}（n+1，\ mathbb {r}）$是非元素fuchsian组$γ$的代表，则与$ρ$相关的单位切线束$ y $与$ρ$相关的单位捆绑包$ y $，由$ρ$定义为affine Group $ b $ b $ b $ b。如果图像$ρ（γ）$满足conze-guivarc'h条件，即强烈的不可还原性和接近性，则$ b $ action的动力学是由$ \ \ \ \ m athbb {rm} {\ rm p}^n $（theerem a）上的$ρ（γ）$的近端动态捕获的。实际上，$ y $的独特$ b $ - 少量子集上的叶状烟节流动的动力学可以用单位切线$ x $ x $ x $ surface $ x $ surface $ s =γ\ backslash \ backslash \ mathbb {h} $（theorem b）的非随机流动的动力学来描述。假设存在连续限制图，我们证明$ b $ - 密西布的套件是叶子霍罗旋转的吸引子，仅限于$ y $（定理C）的非随机设置的近端部分。作为推论，我们推断出限制的流量允许且仅当$γ$ as convex-cocroccact时，限制流动的流动radon的唯一$ u $ u $ u $ u $ invariant ra（定义为乘法常数）。例如，由大炮 - 瑟斯顿图所定义的球体束上的叶状烟节流是独特的终极性。
In this paper we study topological aspects of the dynamics of the foliated horocycle flow on flat projective bundles over hyperbolic surfaces and we derive ergodic consequences. If $ρ: Γ\to {\rm PSL}(n+1,\mathbb{R})$ is a representation of a non-elementary Fuchsian group $Γ$, the unit tangent bundle $Y$ associated to the flat projective bundle defined by $ρ$ admits a natural action of the affine group $B$ obtained by combining the foliated geodesic and horocycle flows. If the image $ρ(Γ)$ satisfies Conze-Guivarc'h conditions, namely strong irreducibility and proximality, the dynamics of the $B$-action is captured by the proximal dynamics of $ρ(Γ)$ on $\mathbb{R}{\rm P}^n$ (Theorem A). In fact, the dynamics of the foliated horocycle flow on the unique $B$-minimal subset of $Y$ can be described in terms of dynamics of the horocycle flow on the non-wandering set in the unit tangent bundle $X$ of the surface $S= Γ\backslash \mathbb{H}$ (Theorem B). Assuming the existence of a continuous limit map, we prove that the $B$-minimal set is an attractor for the foliated horocycle flow restricted to the proximal part of the non-wandering set in $Y$ (Theorem C). As a corollary, we deduce that the restricted flow admits a unique conservative ergodic $U$-invariant Radon measure (defined up to a multiplicative constant) if and only if $Γ$ is convex-cocompact. For example, the foliated horocycle flow on the sphere bundle defined by the Cannon-Thurston map is uniquely ergodic.
