The correlation spectrum for hyperbolic analytic maps.

*(English)*Zbl 0768.58027Iterating a self mapping of a manifold might generate chaotic dynamics. The time correlation function is an appropriate tool for the study of those chaotic systems since it is easy to calculate and may be used as a reliable probe. In order to analyze the correlation function taking the Fourier transform and studying the resulting “correlation spectrum” has been proved to be very useful.

In the present paper, the following theorem is proved: The correlation spectrum associated with analytic observables for a real analytic hyperbolic map on a finite set of rectangles consists of isolated points satisfying exponential bounds. The spectral numbers are given by the reciprocal values of the zeros of a Fredholm determinant.

The class of mappings satisfying the assumptions of this theorem includes certain Hamiltonian systems as well as nonlinear perturbations of the cat map and similar linear Anosov maps.

The paper is organized as follows: First, the time correlation function is defined and the basic notations are introduced. Next, the theorem is proved by use of so-called “pinning” coordinates. A separate section deals with the initial measure which is used for constructing the time correlation function. Finally, a link to Ruelle-Fried theory is established. Examples illustrate the relevant features, in particular, a trivial example is presented along with the definition, ideas and result.

In the present paper, the following theorem is proved: The correlation spectrum associated with analytic observables for a real analytic hyperbolic map on a finite set of rectangles consists of isolated points satisfying exponential bounds. The spectral numbers are given by the reciprocal values of the zeros of a Fredholm determinant.

The class of mappings satisfying the assumptions of this theorem includes certain Hamiltonian systems as well as nonlinear perturbations of the cat map and similar linear Anosov maps.

The paper is organized as follows: First, the time correlation function is defined and the basic notations are introduced. Next, the theorem is proved by use of so-called “pinning” coordinates. A separate section deals with the initial measure which is used for constructing the time correlation function. Finally, a link to Ruelle-Fried theory is established. Examples illustrate the relevant features, in particular, a trivial example is presented along with the definition, ideas and result.

Reviewer: H.Kriete (Aachen)

##### MSC:

37B30 | Index theory for dynamical systems, Morse-Conley indices |

37D30 | Partially hyperbolic systems and dominated splittings |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |