Power spectra for deterministic chaotic dynamical systems

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 Hölder 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.

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