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

TL;DR

This paper investigates the broadband properties of power spectra for a class of intermittent maps with different decay rates of correlations, revealing conditions under which the spectra are continuous and typically nonvanishing.

Contribution

It characterizes the spectral properties of intermittent maps with summable and nonsummable decay of correlations, extending classical results to nonuniformly expanding maps.

Findings

Power spectrum is continuous and nonvanishing for summable decay cases.

In nonsummable decay cases, the spectrum extends continuously and is typically nonzero.

Results apply to a broad class of intermittent maps with specific local behaviors.

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 $\gamma\in(0,\frac12)$, 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 H\"older observables. Moreover, in the nonsummable case $\gamma\in[\frac12,1)$, 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.