Periodic ends, growth rates, Holder dynamics for automorphisms of free groups

Let F-n be the free group of rank n, and partial derivative F-n its boundar y (or space of ends). For any alpha is an element of Aut F-n, the homeomorp hism partial derivative alpha induced by alpha on partial derivative F-n ha s at least two periodic points of period less than or equal to 2n. Periods of periodic points of partial derivative alpha are bounded above by a numbe r M-n, depending only on n, will log M-n similar to root n log n as n --> infinity. Using the canonical Holder structure on partial derivative F-n, w e associate an algebraic number lambda greater than or equal to 1 to any at tracting fixed point X of partial derivative alpha; if lambda > 1, then for any Y close to X the sequence partial derivative alpha(p)(Y) approaches X at about the same speed as e(-lambda P). This leads to a set of Holder expo nents Lambda(h)(Phi) subset of (1, +infinity) associated to any Phi is an e lement of Out F-n. This set coincides with the set of nontrivial exponentia l growth rates of conjugacy classes of F-n under iteration of Phi.