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Broadband nature of power spectra for intermittent maps with summable and nonsummable decay of correlations
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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:10.1088/17518113/49/17/174003
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Official URL: http://dx.doi.org/10.1088/17518113/49/17/174003
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 > Mathematics  
Journal or Publication Title:  Journal of Physics A: Mathematical and Theoretical  
Publisher:  IOP Publishing Ltd  
ISSN:  17518113  
Official Date:  18 March 2016  
Dates: 


Volume:  49  
Number:  17  
Article Number:  174003  
DOI:  10.1088/17518113/49/17/174003  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access 
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