Busemann function and its relation to horocycle Consider Busemann function, $B_{\zeta}(w)$ for the Poincare disk model. Handling with literature, I see that this function relates to the horocycles on the disk. Here (p.148, sec. 9.4.2, def. 9.34) the author just defines horocycle as a level set of Busemann function.
However, this connection is not clear for me. Also, I have a vague feeling that Busemann function somehow relates to Anosov flow.

*

*Can anyone please provide more detailed/clear sources, where the connection between Busemann function and horocycles are discussed?


*Does Anosov flow relate to horocycles/Busemann functions on the Poincare disk?
 A: $\newcommand{\R}{\mathbb{R}}$Here's a quick primer on Busemann functions and horocycles. A Busemann function can be viewed as a (normalized) distance function from a point on the boundary of hyperbolic space, and a horocycle can be viewed as a circle centered at the boundary of hyperbolic space. From this point of view, it makes sense that a horocycle should be the level set of a Busemann function.
Start with a point $p \in H$ and a unit speed geodesic $c: \R \rightarrow H$ such that $c(0) = p$. For each $t$, let $d_t(x)$ be the distance from $c(-t)$ to any point $x \in H$ and $C_t$ be the circle of radius $t$ centered at $c(t)$. In particular, $C_t = \{ x\ :\ d_t(x) = t\}$. Observe that $d_t(p) = t$ and therefore $p \in C_t$.
We basically want to let $t \rightarrow \infty$, but we need to normalize $d_t$ in order for it to converge to a function on $H$. So we define
$$ b_t(x) = d_t(x) - t. $$ Since $b_t$ differs from $d_t$ by only a constant term, its derivatives satisfy the same properties as the derivative of a distance function.
Observe that for any $t \in \R$, $b_t(p) = 0$ and $|\nabla b_t| = |\nabla d_t| = 1$. This implies that, as $t \rightarrow -\infty$, $b_t$ converges uniformly to a continuous function $b$. In fact, all derivatives of $b_t$ are uniformly bounded, so $b_t$ converges smoothly to $b$. $b$ is called a Busemann function. Since the derivatives of $b_t$ have the same properties as the derivatives of a distance function, the same holds for $b$, too. This is why $b$ can be viewed as a normalized distance function.
Meanwhile, the circle $C_t$ has radius $t$, contains $p$, and is the level set $b_t = 0$. Therefore, as $t \rightarrow \infty$, $C_t$ converges smoothly to a curve $C$ containing $p$. It is called a horocycle. The geodesic curvature of circle of radius $t$, such as $C_t$, is $\coth t$ and therefore the geodesic curvature of $C$ is
$$
\lim_{t \rightarrow \infty}\coth t = 1.
$$
