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 ∈ (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] → [0, 1] with f (x) ≈ x 1+γ for x ≈ 0, where γ ∈ (0, 1). Such maps have summable decay of correlations when γ ∈ (0, 1 2 ), 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 ∈ [ 1 2 , 1), we show that S(ω) is defined almost everywhere with a continuous extensionS(ω) defined on (0, 2π), and S(ω) is typically nonvanishing.More generally, the power spectrum is well-defined and continuous provided the autocorrelations are summable.The power spectrum is often used by experimentalists to distinguish periodic and quasiperiodic dynamics (discrete power spectrum with peaks at the harmonics and subharmonics) and chaotic dynamics (broadband power spectra). See for example [10]. In the atmospheric and oceanic sciences and in climate science, power spectra have been widely used to detect variability in particular frequency bands (see, for example, [9,7]). Spectral analysis was successful in detecting dominant time scales in teleconnection patterns, revealing intraseasonal variability in time series of the global atmospheric angular momentum [6], interannual variability in the El Niño/Southern Oscillation system [4,14], and the Atlantic Multidecadal Variability [5]. On millenial temporal scales the power spectrum was instrumental in unraveling dominant cycles in paleoclimatic records [25,26].Despite this widespread applicability across these disparate temporal scales, there are surprisingly few rigorous results on the nature of power spectra of complex systems. The (quasi)periodic case with its peaks at discrete frequencies is well understood. The nature of power spectra for chaotic systems was first treated in [23] in the case of uniformly hyperbolic (Axiom A) systems. In our previous paper [16], we considered in more detail the broadband nature of power spectra for chaotic dynamical systems and showed that for certain classes of dynamical systems f and observables v, the power spectrum is bounded away from zero.The main results in [16] are for nonuniformly expanding/hyperbolic dynamical systems with exponential decay of correlations. These results are summarised below in Subsection 1.1. The current paper is concerned with systems possessing subexponential -even nonsummable -decay of correlations. A prototypical example is the class of Pomeau-Manneville intermittent maps [22], specifically the class considered in [15]. These are maps f : [0, 1] → [0, 1] given by