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Dynamics of the heat semigroup on symmetric spaces. (English) Zbl 1185.37077

Summary: 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 \(1<p\leq 2\). To see this, we show 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 \(1<p\leq 2\). These results are compared with the corresponding situation for Euclidean spaces and symmetric spaces of compact type, where such a behavior is not possible.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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