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Quantitative Quasiperiodicity | Suddhasattwa Das
; Yoshitaka Saiki
; Evelyn Sander
; James A. Yorke
; | Date: |
1 Aug 2015 | Abstract: | The Birkhoff Ergodic Theorem concludes that time averages, that is, Birkhoff
averages, $Sigma_{n=1}^N f(x_n)/N$ of a function $f$ along an ergodic
trajectory $(x_n)$ of a function $T$ converges to the space average $int f
dmu$, where $mu$ is the unique invariant probability measure. Convergence of
the time average to the space average is slow. We introduce a modified average
of $f(x_n)$ by giving very small weights to the "end" terms when $n$ is near
$0$ or $N$. When $(x_n)$ is a trajectory on a quasiperiodic torus and $f$ and
$T$ are $C^infty$, we show that our weighted Birkhoff averages converge
"super" fast to $int f dmu$, {em i.e.} with error smaller than every
polynomial of $1/N$. Our goal is to show that our weighted Birkhoff average is
a powerful computational tool, and this paper illustrates its use for several
examples where the quasiperiodic set is one or two dimensional. In particular,
we compute rotation numbers and conjugacies (i.e. changes of variables) and
their Fourier series, often with 30-digit precision. | Source: | arXiv, 1508.0062 | Other source: | [GID 1682006] 1601.6051 | Services: | Forum | Review | PDF | Favorites |
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