The expansion factors of an outer automorphism and its inverse

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 $\phi^{-1}$. Nevertheless, we prove that the ratio between the logarithms of the expansion factors of $\phi$ and $\phi^{-1}$ 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 two spectrums of expansion factors.