- Transfer operator
: "The transfer operator is different from the transfer homomorphism."
In
mathematics , the transfer operator encodes information about an iterated map and is frequently used to study the behavior ofdynamical systems ,statistical mechanics ,quantum chaos andfractals . The transfer operator is sometimes called the Ruelle operator, afterDavid Ruelle , or the Ruelle-Perron-Frobenius operator in reference to the applicability of theFrobenius-Perron theorem to the determination of the eigenvalues of the operator.The iterated function to be studied is a map f:X ightarrow X for an arbitrary set X. The transfer operator is defined as an operator mathcal{L} acting on the space of functions Phi:X ightarrow mathbb{C} as
:mathcal{L}Phi)(x) = sum_{yin f^{-1}(x)} g(y) Phi(y)
where g:X ightarrowmathbb{C} is an auxiliary valuation function. When f has a
Jacobian determinant, then g is usually taken to be g=1/|J|.Some questions about the form and nature of a transfer operator are addressed in the theory of
composition operator s.The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic
pushforward of "g": in essence, the transfer operator is thedirect image functor in the category of measureable spaces.Applications
Whereas the iteration of a function f naturally leads to a study of the orbits of points of X under iteration (the study of point dynamics), the transfer operator defines how (smooth) maps evolve under iteration. Thus, transfer operators typically appear in
physics problems, such asquantum chaos andstatistical mechanics , where attention is focused on the time evolution of smooth functions.It is often the case that the transfer operator is positive, has discrete positive real-valued
eigenvalue s, with the largest eigenvalue being equalto one. For this reason, the transfer operator is sometimes called the Frobenius-Perron operator.The
eigenfunction s of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantum Hamiltonian, the eigenvalues will typically be very closely spaced, and thus even a very narrow and carefully selected ensemble of quantum states will encompass a large number of very different fractal eigenstates with non-zero support over the entire volume. This can be used to explain many results from classical statistical mechanics, including the irreversibility of time and the increase ofentropy .The transfer operator of the Bernoulli map b(x)=2x-lfloor 2x floor is exactly solvable and is a classic example of deterministic chaos; the discrete eigenvalues correspondto the
Bernoulli polynomials . This operator also has a continuous spectrum consisting of theHurwitz zeta function .The transfer operator of the Gauss map h(x)=1/x-lfloor 1/x floor is called the Gauss-Kuzmin-Wirsing (GKW) operator and due to its extraordinary difficulty, has not been fully solved. The theory of the GKW dates back to a hypothesis by Gauss on
continued fraction s and is closely related to theRiemann zeta function .ee also
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Bernoulli scheme
*Shift of finite type References
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* David Ruelle, " [http://www.maths.ex.ac.uk/~mwatkins/zeta/ruelle.pdf Dynamical Zeta Functions and Transfer Operators] ", (2002) Institut des Hautes Etudes Scientifiques preprint IHES/M/02/66. "(Provides an introductory survey)."
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