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 $S(\omega )$, $\omega \in (0,2\pi )$, 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 $\gamma \in (0,\frac{1}{2})$, 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 $\gamma \in [\frac{1}{2},1)$, 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.

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References in corpus (1)

• Almost Sure Invariance Principle For Nonuniformly Hyperbolic Systems