  A fully irreducible outer automorphism phi of the free group F_n of rank n
has an expansion factor which often differs from the expansion factor of the
inverse of phi. Nevertheless, we prove that the ratio between the logarithms of
the expansion factors of phi and its inverse is bounded above by a constant
depending only on the rank n. We also prove a more general theorem applying to
an arbitrary outer automorphism of F_n and its inverse, and their entire
spectrum of expansion factors.
