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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.

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

  2. Does Anosov flow relate to horocycles/Busemann functions on the Poincare disk?

Artem Alexandrov
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    Ask one big question at a time please, no big questions "in addition". – Lee Mosher Jun 02 '21 at 17:16
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    Also, there are many carefully explained details about horocycles in that section of that book, on the pages following that definition. Much of what I might think of to say, regarding the connection between Busemann functions and horocycles, is right there in those pages. It would greatly improve your question if you could clarify what you do not understand in those pages. – Lee Mosher Jun 02 '21 at 17:23

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$\newcommand{\R}{\mathbb{R}}$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 $$ \lim_{t \rightarrow \infty}\coth t = 1. $$

Deane
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