Operator of transfer

In Mathematical, the operator of transfer encode the information of a reiterated application and is frequently used to study the behavior of the dynamic systems, of the mechanical statistics, the quantum Chaos and the Fractals. The operator of transfer is sometimes called the operator of Lane , in the honor of David Ruelle, or the operator of Lane-Perron-Frobenius referring to the applicability of the Théorème of Frobenius-Perron for the determination of the eigenvalues of the operator.

The studied reiterated function is an application $f: X \ rightarrow X$ of an arbitrary unit $X$ . The operator of transfer is defined like an operator $\ mathcal {L} \,$ acting on the space of the functions $\ Phi: X \ rightarrow \ mathbb {C} \,$ like

$(\ mathcal {L} \ Phi) (X) = \ sum_ {there \ in f^ {- 1} (X)} G (there) \ Phi (there) \,$

where $g: X \ rightarrow \ mathbb {C} \,$ is an auxiliary valuation of function. When $f$ has a Déterminant jacobien, then $g$ is generally taken for $g=1/|J|\,$ .

Certain questions in connection with the shape and the nature of the operator of transfer are studied in the theory of the operators of composition.

Applications

Considering that the iteration of a function

f

leads naturally to the study of the orbits of the points of X under the iteration (the study of the dynamic systems), the operator of transfer defines how the applications (continuous) evolve/move under the iteration. Thus, the operators of transfer frequently appear in the problems of Physique, such as the quantum Chaos and the mechanical statistics, where the attention is concentrated on the temporal evolution of the continuous functions.

The case is frequent where the operator of transfer is positive, eigenvalues has discrete, positive, having actual values, with the greatest eigenvalue being equal to one. For this reason, the operator of transfer is sometimes called the operator of Frobenius-Perron.

The eigenvalues of the operator of transfer are usually fractals. When the logarithm of the operator of transfer corresponds to the quantum Hamiltonien, the eigenvalues typically will be very narrowly aligned, and thus, even a together attentively selected quantum states will be surrounded by a great number of clean states fractals very different with a support different from zero on whole volume. This can be used to explain many results resulting from traditional statistical mechanics, including the irreversability of time as well as the increase in the Entropie.

The operator of transfer of the application of Bernoulli $b (X) =2x- \ lfloor 2x \ rfloor \,$ is résolvable exactly and is a traditional example of deterministic chaos; the discrete eigenvalues correspond to the polynôes of Bernoulli. This operator has also a continuous spectrum constituting the Fonction zeta of Hurwitz.

The operator of transfer of the application of Gauss $h (X) =1/x- \ lfloor 1/x \ rfloor \,$ is called the operator of Gauss-Kuzmin-Wirsing (GKW) and because of its extraordinary difficulty, was not fully solved yet. The theory of operator GKW goes up with the assumption made per Gauss on the continued fractions and is connected firmly to the Fonction zeta of Riemann.

See too

Diagram of Bernoulli

References

David Ruelle, Thermodynamic formalism: the mathematical structures off classical equilibrium statistical mechanics

Dieter H. Mayer, The Lane-Araki transfer operator in classical statistical mechanics

David Ruelle, Dynamical Zeta Functions and Transfer Operators , (2002) Institute of the High Scientific studies preprint IHES/M/02/66. (Provides year introductory survey).

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