  The inverse of the fermion matrix squared is used to define a transfer matrix
for domain-wall fermions. When the domain-wall height $M$ is bigger than one,
the transfer matrix is complex. Slowly suppressed chiral symmetry violations
may then arise from all eigenvalues of the transfer matrix which are located
near the unit circle. Using a variable lattice spacing for the fifth coordinate
we enforce the strict positivity of the transfer matrix for any $M$. We
furthermore propose a modified pseudo-fermion action, aimed to decrease the
density of close-to-unity eigenvalues of the (positive) transfer matrix, at the
price of a small renormalization of the coupling constant. We explain why these
changes may reduce chiral symmetry violations in lattice QCD simulations.
