Gottwald, Georg A. and Melbourne, Ian (2016) Broadband nature of power spectra for intermittent maps with summable and nonsummable decay of correlations. Journal of Physics A: Mathematical and Theoretical, 49 (17). 174003. doi:10.1088/1751-8113/49/17/174003 ISSN 1751-8113.

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Abstract

We present results on the broadband nature of the power spectrum $S(\omega )$, $\omega \in (0,2\pi )$, for a large class of nonuniformly expanding maps with summable and nonsummable decay of correlations. In particular, we consider a class of intermittent maps $f\;:[0,1]\to [0,1]$ with $f(x)\approx {x}^{1+\gamma }$ for $x\approx 0$, where $\gamma \in (0,1)$. Such maps have summable decay of correlations when $\gamma \in \left(0,\frac{1}{2}\right)$, and $S(\omega )$ extends to a continuous function on $[0,2\pi ]$ by the classical Wiener–Khintchine theorem. We show that $S(\omega )$ is typically bounded away from zero for Hölder observables. Moreover, in the nonsummable case $\gamma \in \left[\frac{1}{2},1\right)$, we show that $S(\omega )$ is defined almost everywhere with a continuous extension $\tilde{S}(\omega )$ defined on $(0,2\pi )$, and $\tilde{S}(\omega )$ is typically nonvanishing.

Item Type:	Journal Article
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Journal of Physics A: Mathematical and Theoretical
Publisher:	IOP Publishing Ltd
ISSN:	1751-8113
Official Date:	18 March 2016
Dates:	Date Event 18 March 2016 Published
Volume:	49
Number:	17
Article Number:	174003
DOI:	10.1088/1751-8113/49/17/174003
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

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