On the spectrum of shear flows and uniform ergodic theorems. (English) Zbl 1294.35044
Summary: The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both \(L^2\) spaces and weighted-\(L^2\) spaces. As a consequence, an example of a flow admitting a purely singular continuous spectrum is provided. For flows admitting more regular spectra the density of states is analyzed, and spaces on which it is uniformly bounded are identified. As an application, an ergodic theorem with uniform convergence is proved.
MSC:
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
| 47A35 | Ergodic theory of linear operators |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
Keywords:
first order differential operator; density of states; uniform ergodic theorem; singular continuous spectrumReferences:
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