Broadband nature of power spectra for intermittent Maps with summable and nonsummable decay of correlations

Abstract

We present results on the broadband nature of the power spectrum $S(ω)$, $ω\in(0,2π)$, 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 $f(x)\approx x^{1+γ}$ for $x\approx 0$, where $γ\in(0,1)$. Such maps have summable decay of correlations when $γ\in(0,\frac12)$, and $S(ω)$ extends to a continuous function on $[0,2π]$ by the classical Wiener-Khintchine Theorem. We show that $S(ω)$ is typically bounded away from zero for Hölder observables. Moreover, in the nonsummable case $γ\in[\frac12,1)$, we show that $S(ω)$ is defined almost everywhere with a continuous extension $\tilde S(ω)$ defined on $(0,2π)$, and $\tilde S(ω)$ is typically nonvanishing.