Let $G$ be the group of isometries of the Poincaré disk. Let $\mu$ be a probability measure on $G$, and consider $g_1,..,g_n$ i.i.d. random variables on $G$ distributed according to $\mu$. For $z\in \mathbb{C}, |z|<1$, I would like to understand what can be said about the sequence

$$ z_n = g_n...g_1 (z),\quad z\in \mathbb{C}, |z|<1. $$

In particular, I wish to get an understanding of:
-when does $d(0,z_n)$ goes to infinity
-when it does, what is the behavior of $z_n$ as it gets closer to the boundary (does it converges?)

I feel that it is a standard question but I couldn't find yet a result which corresponds precisely to this setting. Could anyone help? Thank you.

  • 3
    $\begingroup$ What do you assume on $\mu$? You don't even say it's a probability... maybe you assume even more? a plain Borel probability measure? Radon? or you might allow finitely supported? There are many results about this question, and they might include some finite moment assumption. (Search [Random walk hyperbolic plane] yields a lot of answers.) $\endgroup$
    – YCor
    Mar 25, 2021 at 15:18
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    $\begingroup$ The order of composition has to be $g_1\cdots g_n z$ for there to be a chance of convergence. For example, imagine that $\mu$ gives positive weight to some rotation fixing $0$. Then with positive frequency $g_n$ will equal said rotation and this means $g_n\cdots g_1(z)$ cannot converge to a point on the boundary. $\endgroup$ Mar 25, 2021 at 15:55
  • 2
    $\begingroup$ Another relevant keyword is "Poisson boundary" $\endgroup$
    – YCor
    Mar 25, 2021 at 16:05
  • 4
    $\begingroup$ Under quite general conditions on $\mu$, the positivity of the speed $\lim \frac{1}{n}d(z_0,z_n)$ follows from Furstenberg's criteria for positivity of the first Lyapunov exponent (you need to know how to translate this to a statement about a product of i.i.d. matrices, this is explained in any book about plane hyperbolic geometry). The original reference is Furstenberg's 1963 article "Non commuting random products", in particular Section 5 discusses application to a setting including yours (this article is very dense, but worth looking at). $\endgroup$ Mar 25, 2021 at 16:10
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    $\begingroup$ For convergence to the boundary. In the hyperbolic plane any sequence with $\liminf \frac{1}{n}d(z_0,z_n) > 0$ and $\lim \frac{1}{n}d(z_n,z_{n+1}) = 0$ converges to a point on the boundary. You can prove this via trigonometry. More general (variable curvature) versions of this idea appear, for example, in Kaimanovich's proof of Oseledets theorem from the late 70's, and in Karlsson and Margulis' 1999 "Multiplicative ergodic theorems...". For curves in hyperbolic space you can look at Boyland's 2000 paper "New Dynamical invariants" Lemma 2.1. $\endgroup$ Mar 25, 2021 at 16:32


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