Inmathematics, thetransfer operator encodes information about aniterated map and is frequently used to study the behavior ofdynamical systems,statistical mechanics,quantum chaos andfractals. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is theinvariant measure of the system.

The transfer operator is sometimes called theRuelle operator, afterDavid Ruelle, or thePerron–Frobenius operator orRuelle–Perron–Frobenius operator, in reference to the applicability of thePerron–Frobenius theorem to the determination of theeigenvalues of the operator.

The iterated function to be studied is a map${\displaystyle f:X\to X}$ for an arbitrary set${\displaystyle X}$.

The transfer operator is defined as an operator${\displaystyle \mathcal{L}}$ acting on the space of functions${\displaystyle \{\mathrm{\Phi }:X\to \mathbb{C}\}}$ as

${\displaystyle (\mathcal{L}\mathrm{\Phi })(x)=\sum _{y\in f^{-1}(x)}g(y)\mathrm{\Phi }(y)}$

where${\displaystyle g:X\to \mathbb{C}}$ is an auxiliary valuation function. When${\displaystyle f}$ has aJacobian determinant${\displaystyle |J|}$, then${\displaystyle g}$ is usually taken to be${\displaystyle g=1/|J|}$.

The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoreticpushforward ofg: in essence, the transfer operator is thedirect image functor in the category ofmeasurable spaces. The left-adjoint of the Perron–Frobenius operator is theKoopman operator orcomposition operator. The general setting is provided by theBorel functional calculus.

As a general rule, the transfer operator can usually be interpreted as a (left-)shift operator acting on ashift space. The most commonly studied shifts are thesubshifts of finite type. The adjoint to the transfer operator can likewise usually be interpreted as a right-shift. Particularly well studied right-shifts include theJacobi operator and theHessenberg matrix, both of which generate systems oforthogonal polynomials via a right-shift.

Whereas the iteration of a function${\displaystyle f}$ naturally leads to a study of the orbits of points of X under iteration (the study ofpoint dynamics), the transfer operator defines how (smooth) maps evolve under iteration. Thus, transfer operators typically appear inphysics problems, such asquantum chaos andstatistical mechanics, where attention is focused on the time evolution of smooth functions. In turn, this has medical applications torational drug design, through the field ofmolecular dynamics.

It is often the case that the transfer operator is positive, has discrete positive real-valuedeigenvalues, with the largest eigenvalue being equal to one. For this reason, the transfer operator is sometimes called the Frobenius–Perron operator.

Theeigenfunctions of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantumHamiltonian, the eigenvalues will typically be very closely spaced, and thus even a very narrow and carefully selectedensemble of quantum states will encompass a large number of very different fractal eigenstates with non-zerosupport 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 theBernoulli map${\displaystyle b(x)=2x-\lfloor 2x\rfloor }$ is exactly solvable and is a classic example ofdeterministic chaos; the discrete eigenvalues correspond to theBernoulli polynomials. This operator also has a continuous spectrum consisting of theHurwitz zeta function.

The transfer operator of the Gauss map${\displaystyle h(x)=1/x-\lfloor 1/x\rfloor }$ is called theGauss–Kuzmin–Wirsing (GKW) operator. The theory of the GKW dates back to a hypothesis by Gauss oncontinued fractions and is closely related to theRiemann zeta function.

• Bernoulli scheme
• Shift of finite type
• Krein–Rutman theorem
• Transfer-matrix method

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• Gaspard, Pierre (1998).Chaos, scattering and statistical mechanics.Cambridge University Press.ISBN 0-521-39511-9.
• Mackey, Michael C. (1992).Time's Arrow : The origins of thermodynamic behaviour. Springer-Verlag.ISBN 0-387-94093-6.
• Mayer, Dieter H. (1978).The Ruelle-Araki transfer operator in classical statistical mechanics. Springer-Verlag.ISBN 0-387-09990-5.
• Ruelle, David (1978).Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics. Addison–Wesley, Reading.ISBN 0-201-13504-3.
• Ruelle, David (2002)."Dynamical Zeta Functions and Transfer Operators"(PDF).Notices of the AMS.49 (8):887–895.(Provides an introductory survey).

• Chaos theory
• Dynamical systems
• Operator theory
• Spectral theory