A Fréchet law and an Erdős-Philipp law for maximal cuspidal windings (Q2842224)

Let \(\mathbb{H}\) denote the upper half plane model of the real hyperbolic plane, and let \(\mathbb{H}(\infty) = \mathbb{R} \cup \{ \infty\}\) denote the circle of points at infinity for \(\mathbb{H}\). Let \(G\) denote an essentially free, finitely generated Fuchsian group that contains parabolic elements and operates by isometries on \(\mathbb{H}\).NEWLINENEWLINEThe limit set \(L(G) \subset \mathbb{H}(\infty)\) is the \(G\)-invariant set of accumulation points of the orbit \(G(i), i = (0,1) \in \mathbb{H}\). The radial limit set \(L_{r}(G)\) consists of those points \(\xi \in L(G)\) such that the corresponding geodesic \(\gamma_{\xi}\) starting from \(i\) satisfies \(d(\gamma_{\xi}(t_{n}), g_{n}(i)) \leq c\) for some positive number \(c\) and some sequences \(\{g_{n} \} \subset G\) and \(\{t_{n} \} \subset (0, \infty)\) with \(t_{n} \rightarrow + \infty\) as \(n \rightarrow + \infty\). If \(\xi \in L_{r}(G)\), then the projection of \(\gamma_{\xi}\) into \(\mathbb{H} / G\) cannot remain in any given cusp of \(\mathbb{H} / G\) for an interval \([T, \infty)\), with \(T > 0\). Moreover, if \(D\) denotes the Dirichlet fundamental domain of \(G\) with center at \(i\), then \(\gamma_{}[0,\infty)\) meets infinitely many copies \(g(D)\), \(g \in G\).NEWLINENEWLINEOne defines the critical exponent \(\delta(G) = \lim_{n \rightarrow \infty} \frac{1}{n} \log |\{g \in G : d(i,g(i) \leq n \}|\). It is known that \(\delta(G) > \frac{1}{2}\) since \(G\) has parabolic elements. From \(\delta(G)\) one obtains a Patterson probability measure \(m_{\delta}\) on \(\mathbb{H}(\infty)\) that is supported on \(L(G)\).NEWLINENEWLINE\(G\) is known to be a free product \(H*\Gamma\), where \(H\) is a free product of finitely many elementary hyperbolic subgroups \(\langle h_{i}, h_{i}^{-1} \rangle , 1 \leq i \leq N\), and \(\Gamma\) is a free product of finitely many elementary parabolic subgroups \(\Gamma_{j} = \langle \gamma_{j}, \gamma_{j}^{-1} \rangle , 1 \leq j \leq M\). Every element of \(G\) is an infinite word in the elements of \(G_{0} = \{ h_{i}, h_{i}^{-1}, \gamma_{j}, \gamma_{j}^{-1} : 1 \leq i \leq N, 1 \leq j \leq M\}\). Moreover, every element \(\xi \in L_{r}(G)\) defines an infinite reduced word \(w_{\xi}\) in the elements of \(G_{0}\) that records in succession the copies \(g(D)\) of the fundamental domain \(D\) through which the geodesic \(\gamma_{\xi}\) passes.NEWLINENEWLINEThe word \(w_{\xi}\) can be decomposed into a sequence of blocks. A hyperbolic element \(g \in w_{\xi}\) defines a block that contains only \(g\). A hyperbolic block has length 1. A parabolic element \(g \in w_{\xi}\) defines a block in which \(g\) appears \(n\) times consecutively in \(w_{\xi}\) but not \(n+1\) times. Such a parabolic block has length \(n\). The geometric meaning of the integer \(n\) is that the projection of the geodesic \(\gamma_{\xi}\) into \(\mathbb{H} / G\) spirals \(n-1\) times around the cusp of \(\mathbb{H} / G\) determined by \(g\) before exiting the cusp.NEWLINENEWLINEFor \(\xi \in L_{r}(G)\) define \(X_{k}(\xi)\) to be the length of the \(k\)th block of \(w_{\xi}\), and for each positive integer \(n\) define \(Y_{n}(\xi) = \max\{X_{1}(\xi),\dots, X_{n}(\xi) \}\). The authors prove two main results and give applications and relationships to the work of others.NEWLINENEWLINETheorem 1. Let \(G\) be as above with critical exponent \(\delta(G)\) and corresponding Patterson measure \(m_{\delta}\) on \(L(G)\). Let \(\nu\) be a probability measure on \(L(G)\) that is absolutely continuous with respect to \(m_{\delta}\). Then for each \(s > 0\), \(\lim_{n \rightarrow \infty} \nu(\{\xi \in L_{r}(G) : Y_{n}(\xi)^{2 \delta -1} / n \leq s \}) = \exp(-\kappa(G) / s)\), where \(\kappa(G)\) is an explicitly defined positive constant that depends only on \(G\).NEWLINENEWLINETheorem 2. Let \(G\), \(\delta\) and \(m_{\delta}\) be as above. Then there exists a full measure subset \(A\) of \(L_{r}(G)\) such that \(\liminf_{n \rightarrow \infty} Y_{n}(\xi)^{2 \delta - 1} (\log\log n) / n = \kappa(G)\) for all \(\xi \in A\). Moreover, for any sequence \(\{\ell_{n} \} \subset (0,\infty)\) one has \(\limsup_{n \rightarrow \infty}Y_{n} / \ell_{n} = 0\) or \(\infty\).