# zbMATH — the first resource for mathematics

Periodic ends, growth rates, Hölder dynamics for automorphisms of free groups. (English) Zbl 0965.20026
Nielsen analyzed the induced action of a surface homeomorphism, or equivalently of an automorphism of its fundamental group, on the sphere at infinity of the universal covering of the surface; in particular he showed that there are at least two periodic points on the sphere at infinity. In the present paper, in a similar spirit the action $$\partial\alpha$$ of an automorphism $$\alpha$$ of a free group $$F_n$$ of rank $$n$$ on the boundary $$\partial F_n$$ of the free group is analyzed (the Gromov boundary or space of ends which is a Cantor set). Using $$\alpha$$-invariant $$\mathbb{R}$$-trees, it is shown that the action of $$\partial\alpha$$ on $$\partial F_n$$ has at least two periodic points of period $$\leq 2n$$ (typically two fixed points and no other periodic points), and that the period of a periodic point of $$\partial\alpha$$ is bounded above by a constant depending only on $$n$$ which behaves asymptotically like $$\exp\sqrt{n\log n}$$ as $$n\to\infty$$ (this is asymptotically also the maximum order of torsion elements in the automorphism group of $$F_n$$). Also, using the canonical Hölder structure on $$\partial F_n$$, an algebraic number is associated to each attracting fixed point of $$\partial\alpha$$ which determines the local dynamics of $$\partial\alpha$$ around the fixed point. This leads to a set of Hölder exponents associated to any outer automorphism of $$F_n$$ which coincides with the set of nontrivial exponential growth rates of conjugacy classes of $$F_n$$ under iteration of the outer automorphism.

##### MSC:
 20F65 Geometric group theory 20E05 Free nonabelian groups 20E36 Automorphisms of infinite groups 57M05 Fundamental group, presentations, free differential calculus 37B10 Symbolic dynamics
Full Text: