Countable oscillator systems are non-wandering, continuous systems with absolutely continuous measures are wandering, and a Fourier condition extends wandering to a class of singular Bernoulli measures.
An Infinite-dimensional KAM Theorem with Normal Degeneracy
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abstract
In this paper, we consider a classical Hamiltonian normal form with degeneracy in normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. In stead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to be verified, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.
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Measures and Trajectory Properties in Oscillator Systems
Countable oscillator systems are non-wandering, continuous systems with absolutely continuous measures are wandering, and a Fourier condition extends wandering to a class of singular Bernoulli measures.