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The aim of this paper is to show that the dynamics of L p heat semigroups (p > 2) on a symmetric space of non-compact type is very different from the dynamics of the L p heat semigroups if 1 < p ≤ 2. To see this, it is shown that certain shifts of the L p heat semigroups have a chaotic behavior if p > 2, and that such a behavior is not possible in the cases 1 < p ≤ 2. These results are compared with the corresponding situation for euclidean spaces and symmetric spaces of compact type, where such a behavior is not possible.