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

Large supremum norms and small Shannon entropy for Hecke eigenfunctions of quantized cat maps