Periodic ends, growth rates, Hölder dynamics for automorphisms of free groups (Q1589755)

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.