  Geodesic motion determines important features of spacetimes. Null unstable
geodesics are closely related to the appearance of compact objects to external
observers and have been associated with the characteristic modes of black
holes. By computing the Lyapunov exponent, which is the inverse of the
instability timescale associated with this geodesic motion, we show that, in
the eikonal limit, quasinormal modes of black holes in any dimensions are
determined by the parameters of the circular null geodesics. This result is
independent of the field equations and only assumes a stationary, spherically
symmetric and asymptotically flat line element, but it does not seem to be
easily extendable to anti-de Sitter spacetimes. We further show that (i) in
spacetime dimensions greater than four, equatorial circular timelike geodesics
in a Myers-Perry black hole background are unstable, and (ii) the instability
timescale of equatorial null geodesics in Myers-Perry spacetimes has a local
minimum for spacetimes of dimension d > 5.
