Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms

Bowen, Rufus

doi:10.1007/bfb0081279

Cited by 2,086 publications

(2,203 citation statements)

References 36 publications

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“…This is achieved by making obvious modifications to the arguments in [20,3]. First, for each x in X, choosex ∈ X such that the operation x →x depends only on future coordinates and (x) n = x n for all n ≥ 0.…”

Section: Axiom a Systemsmentioning

confidence: 99%

“…Given any Hölder potential on X, there is a unique equilibrium measure µ, and µ|X i is mixing under f q . Again it is well-known [3] that f q |X i has exponential decay of correlations for Hölder observables. Our results extend to a large class of nonuniformly hyperbolic systems modelled by the tower construction of Young [23,24].…”

Section: Introductionmentioning

confidence: 99%

“…(S(2πk/q) = +∞ if and only if v k = 0.) Moreover,S(ω) = 0 if and only if the criterion (2.3) is satisfied for some continuous function χ : X → C. [3,17].…”

Section: Introductionmentioning

confidence: 99%

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Power spectra for deterministic chaotic dynamical systems

Melbourne

Gottwald

2007

Nonlinearity

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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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Section: Axiom a Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Power spectra for deterministic chaotic dynamical systems

Melbourne

Gottwald

2007

Nonlinearity

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“…[10], [2], [16]) our method uses thermodynamic formalism (see, for example, [6], [20]). Let M be the universal cover of X.…”

Section: Estimates For Negatively Curved Manifoldsmentioning

confidence: 99%

Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds

Jakobson

Polterovich²

2005

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We announce asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in the local Weyl's law on Riemannian manifolds. In the negatively curved case, methods of thermodynamic formalism are applied to improve the estimates. Our results develop and extend the unpublished thesis of A. Karnaukh. We discuss some ideas of the proofs; for complete proofs see our extended paper on the subject.

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“…, e n } is the edge set of X with a given direction, and k is a positive integer. For each edge e i ∈ E, we can give e i the partition {I (k) i,j }, 1 ≤ j ≤ j(i, k), for f k such that (1) the initial point of I (k) i,1 is the initial point of e i , (2) the terminal point of I (k) i,j is the initial point of I (k) i,j+1 for 1 ≤ j < j(i, k), (3) the terminal point of I (k) i,j (i,k) is the terminal point of e i , (4) …”

mentioning

confidence: 99%

Canonical symbolic dynamics for one-dimensional generalized solenoids

2001

Trans. Amer. Math. Soc.

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Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms

Cited by 2,086 publications

References 36 publications

Power spectra for deterministic chaotic dynamical systems

Power spectra for deterministic chaotic dynamical systems

Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds

Canonical symbolic dynamics for one-dimensional generalized solenoids

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