  We study black branes carrying both electric and magnetic charges in
Einstein-Maxwell theory coupled to a dilaton-axion in asymptotically anti de
Sitter space. After reviewing and extending earlier results for the case of
electrically charged branes, we characterise the thermodynamics of magnetically
charged branes. We then focus on dyonic branes in theories which enjoy an
$SL(2,R)$ electric-magnetic duality. Using $SL(2,R)$, we are able to generate
solutions with arbitrary charges starting with the electrically charged
solution, and also calculate transport coefficients. These solutions all
exhibit a Lifshitz-like near-horizon geometry. The system behaves as expected
for a charged fluid in a magnetic field, with non-vanishing Hall conductance
and vanishing DC longitudinal conductivity at low temperatures. Its response is
characterised by a cyclotron resonance at a frequency proportional to the
magnetic field, for small magnetic fields. Interestingly, the DC Hall
conductance is related to the attractor value of the axion. We also study the
attractor flows of the dilaton-axion, both in cases with and without an
additional modular-invariant scalar potential. The flows exhibit intricate
behaviour related to the duality symmetry. Finally, we briefly discuss
attractor flows in more general dilaton-axion theories which do not enjoy
$SL(2,R)$ symmetry.
