  This paper concerns the behavior of eigenfunctions of quantized cat maps and
in particular their supremum norm. We observe that for composite integer values
of N, the inverse of Planck's constant, some of the desymmetrized
eigenfunctions have very small support and hence very large supremum norm. We
also prove an entropy estimate and show that our functions satisfy equality in
this estimate. In the case when N is a prime power with even exponent we
calculate the supremum norm for a large proportion of all desymmetrized
eigenfunctions and we find that for a given N there is essentially at most four
different values these assume.
