  A linear automorphism of Euclidean space is called bi-circular its
eigenvalues lie in the disjoint union of two circles $C_1$ and $C_2$ in the
complex plane where the radius of $C_1$ is $r_1$, the radius of $C_2$ is $r_2$,
and $0 < r_1 < 1 < r_2$. A well-known theorem of Philip Hartman states that a
local $C^{1,1}$ diffeomorphism $T$ of Euclidean space with a fixed point $p$
whose derivative $DT_p$ is bi-circular is $C^{1,\beta}$ linearizable near $p$.
We generalize this result to $C^{1,\alpha}$ diffeomorphisms $T$ where $0 <
\alpha < 1$. We also extend the result to local diffeomorphisms in Banach
spaces with $C^{1,\alpha}$ bump functions. The results apply to give simpler
proofs under weaker regularity conditions of classical results of L. P.
Shilnikov on the existence of horseshoe dynamics near so-called saddle-focus
critical points of vector fields in $R^3$.
