J. Phys. A 49 (2016) 174003 (17 pages)
Georg Gottwald and Ian Melbourne
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)≈ x1+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 extension S0(ω) defined on (0,2π), and S0(ω) is typically nonvanishing.