Most automorphisms of a hyperbolic group have very simple dynamics

Let G br a non-elementary hyperbolic group (e.g. a non-abelian free group o f finite rank;). We show that, for "most" automorphisms a of G (in a precis e sense), there exist distinct elements X+, X- the Gromov boundary partial derivative G of G such that lim(n-->+infinity) alpha(+/-n) (g) = X+/- for e very g epsilon G which is not periodic under alpha. This follows from the f act that the homeomorphism partial derivative alpha induced on partial deri vative G has North-South (loxodromic) dynamics. (C) 2000 Editions scientifi ques et medicales Elsevier SAS.