Broadband Nature of Power Spectra for Intermittent Maps with Summable and Nonsummable Decay of Correlations

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) ≈ x1+γ for x ≈ 0, where γ ∈ (0, 1). Such maps have summable decay of correlations when γ ∈ (0, 12), and S(ω) extends to a continuous function on [0, 2π] by the classical WienerKhintchine Theorem. We show that S(ω) is typically bounded away from zero for Hölder observables. Moreover, in the nonsummable case γ ∈ [12 , 1), we show that S(ω) is defined almost everywhere with a continuous extension S̃(ω) defined on (0, 2π), and S̃(ω) is typically nonvanishing.