  We investigate the asymptotic properties of inertial modes confined in a
spherical shell when viscosity tends to zero. We first consider the mapping
made by the characteristics of the hyperbolic equation (Poincar\'e's equation)
satisfied by inviscid solutions. Characteristics are straight lines in a
meridional section of the shell, and the mapping shows that, generically, these
lines converge towards a periodic orbit which acts like an attractor.
  We then examine the relation between this characteristic path and
eigensolutions of the inviscid problem and show that in a purely
two-dimensional problem, convergence towards an attractor means that the
associated velocity field is not square-integrable. We give arguments which
generalize this result to three dimensions. We then consider the viscous
problem and show how viscosity transforms singularities into internal shear
layers which in general betray an attractor expected at the eigenfrequency of
the mode. We find that there are nested layers, the thinnest and most internal
layer scaling with $E^{1/3}$-scale, $E$ being the Ekman number. Using an
inertial wave packet traveling around an attractor, we give a lower bound on
the thickness of shear layers and show how eigenfrequencies can be computed in
principle. Finally, we show that as viscosity decreases, eigenfrequencies tend
towards a set of values which is not dense in $[0,2\Omega]$, contrary to the
case of the full sphere ($\Omega$ is the angular velocity of the system).
Hence, our geometrical approach opens the possibility of describing the
eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers
($10^{-10}-10^{-20}$), which are out of reach numerically, and this for a wide
class of containers.

Fluid Dynamics;Astrophysics;Condensed Matter;Pattern Formation and Solitons