Periodic ends, growth rates, H�lder dynamics for automorphisms of free groups

Abstract

Let F n be the free group of rank n, and \( \partial F_n \) its boundary (or space of ends).¶For any \( \alpha \in \) Aut F n , the homeomorphism \( \partial \alpha \) induced by \( \alpha \) on \( \partial F_n \) has at least two periodic points of period \( \le 2n \). Periods of periodic points of \( \partial \alpha \) are bounded above by a number M n depending only on n, with log \( M_n \sim \sqrt{n\log n} \) as \( n \to + \infty \).¶Using the canonical Holder structure on \( \partial F_n \), we associate an algebraic number \( \lambda \ge 1 \) to any attracting fixed point X of \( \partial \alpha \); if \( \lambda > 1 \), then for any Y close to X the sequence \( \partial \alpha^{p}(Y) \) approaches X at about the same speed as \( e^{-\lambda ^p} \). This leads to a set of Holder exponents \( \Lambda_{h}(\Phi) \subset (1,+\infty) \) associated to any \( \Phi \in \) Out F n . This set coincides with the set of nontrivial exponential growth rates of conjugacy classes of F n under iteration of \( \Phi \).