Nonlinearity 21 (2008) 179-189.
Georg Gottwald and Ian Melbourne
Abstract
We present results on the broadband nature of power spectra for large classes of discrete chaotic dynamical systems, including uniformly hyperbolic (Axiom A) diffeomorphisms and certain nonuniformly hyperbolic diffeomorphisms (such as the Hénon map). Our results also apply to noninvertible maps, including Collet-Eckmann maps. For such maps (even the nonmixing ones) and H&omul;lder continuous observables, we prove that the power spectrum is analytic except for finitely many removable singularities, and that for typical observables the spectrum is nowhere zero. Indeed, we show that the power spectrum is bounded away from zero except for infinitely degenerate observables.
For slowly mixing systems such as Pomeau-Manneville intermittency maps,
where the power spectrum is at most finitely differentiable, nonvanishing
of the spectrum remains valid provided the decay of correlations is summable.
Correction
Section 4 contains a sketch of an argument to show that nonuniformly expanding maps with summable decay of correlations have nonvanishing power spectra. Our later paper points out (see Remark 1.4 therein) that there is a gap in the argument and gives a correct proof. (The later paper also considers the case of nonsummable decay of correlations.) The main parts of the current paper (for uniformly expanding maps and for Axiom A diffeomorphisms) are unaffected by this error.