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Pure point spectrum for dynamical systems and mean, Besicovitch and Weyl almost periodicity
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 2 / February 2024
- Published online by Cambridge University Press:
- 27 March 2023, pp. 524-568
- Print publication:
- February 2024
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- Article
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We consider metrizable ergodic topological dynamical systems over locally compact,
$\sigma $-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More specifically, we characterize pure point spectrum via mean almost periodicity of generic points. We then go on and show how Besicovitch almost periodic points determine both eigenfunctions and the measure in this case. After this, we characterize those systems arising from Weyl almost periodic points and use this to characterize weak and Bohr almost periodic systems. Finally, we consider applications to aperiodic order.
