spectrum dichotomy
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2014 ◽  
Vol 35 (8) ◽  
pp. 2587-2612 ◽  
Author(s):  
JIAN LI ◽  
SIMING TU ◽  
XIANGDONG YE

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy.


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