# Transfer operator

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{{Short description|Operator encoding information about iterated map}}
{{distinguish|transfer homomorphism}}

In [mathematics](/source/mathematics), the '''transfer operator''' encodes information about an [iterated map](/source/iterated_map) and is frequently used to study the behavior of [dynamical systems](/source/dynamical_systems), [statistical mechanics](/source/statistical_mechanics), [quantum chaos](/source/quantum_chaos) and [fractals](/source/fractals). In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the [invariant measure](/source/invariant_measure) of the system.

The transfer operator is sometimes called the '''Ruelle operator''', after [David Ruelle](/source/David_Ruelle), or the '''Perron–Frobenius operator''' or '''Ruelle–Perron–Frobenius operator''', in reference to the applicability of the [Perron–Frobenius theorem](/source/Perron%E2%80%93Frobenius_theorem) to the determination of the [eigenvalue](/source/eigenvalue)s of the operator.

==Definition==
The iterated function to be studied is a map <math>f\colon X\rightarrow X</math> for an arbitrary set <math>X</math>. 

The transfer operator is defined as an operator <math>\mathcal{L}</math> acting on the space of functions <math>\{\Phi\colon X\rightarrow \mathbb{C}\}</math> as

:<math>(\mathcal{L}\Phi)(x) = \sum_{y\,\in\, f^{-1}(x)} g(y) \Phi(y)</math>

where <math>g\colon X\rightarrow\mathbb{C}</math> is an auxiliary valuation function. When <math>f</math> has a [Jacobian](/source/Jacobian_matrix_and_determinant) determinant <math>|J|</math>, then <math>g</math> is usually taken to be <math>g=1/|J|</math>.

The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic [pushforward](/source/Pushforward_measure) of ''g'': in essence, the transfer operator is the [direct image functor](/source/direct_image_functor) in the category of [measurable space](/source/measurable_space)s.  The left-adjoint of the Perron&ndash;Frobenius operator is the [Koopman operator](/source/Koopman_operator) or [composition operator](/source/composition_operator). The general setting is provided by the [Borel functional calculus](/source/Borel_functional_calculus).

As a general rule, the transfer operator can usually be interpreted as a (left-)[shift operator](/source/shift_operator) acting on a [shift space](/source/shift_space). The most commonly studied shifts are the [subshifts of finite type](/source/subshifts_of_finite_type). The adjoint to the transfer operator can likewise usually be interpreted as a right-shift. Particularly well studied right-shifts include the [Jacobi operator](/source/Jacobi_operator) and the [Hessenberg matrix](/source/Hessenberg_matrix), both of which generate systems of [orthogonal polynomials](/source/orthogonal_polynomials) via a right-shift.

==Applications==
Whereas the iteration of a function <math>f</math> naturally leads to a study of the orbits of points of X under iteration (the study of [point dynamics](/source/Chaos_theory)), the transfer operator defines how (smooth) maps evolve under iteration.  Thus, transfer operators typically appear in  [physics](/source/physics) problems, such as [quantum chaos](/source/quantum_chaos) and [statistical mechanics](/source/statistical_mechanics), where attention is focused on the time evolution of smooth functions. In turn, this has medical applications to [rational drug design](/source/rational_drug_design), through the field of [molecular dynamics](/source/molecular_dynamics).

It is often the case that the transfer operator is positive, has discrete positive real-valued [eigenvalue](/source/eigenvalue)s, with the largest eigenvalue being equal to one.  For this reason, the transfer operator is sometimes called the  Frobenius&ndash;Perron operator.

The [eigenfunction](/source/eigenfunction)s of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantum [Hamiltonian](/source/Hamiltonian_(quantum_theory)), the eigenvalues will typically be very closely spaced, and thus even a very narrow and carefully selected [ensemble](/source/quantum_ensemble) of quantum states will encompass a large number of very different fractal eigenstates with non-zero [support](/source/support_(mathematics)) over the entire volume.  This can be used to explain many results from classical statistical mechanics, including the irreversibility of time and the increase of [entropy](/source/entropy).

The transfer operator of the [Bernoulli map](/source/Bernoulli_map) <math>b(x)=2x-\lfloor 2x\rfloor</math> is exactly solvable and is a classic example of [deterministic chaos](/source/chaos_theory); the discrete eigenvalues correspond to the [Bernoulli polynomials](/source/Bernoulli_polynomials).  This operator also has a continuous spectrum consisting of the [Hurwitz zeta function](/source/Hurwitz_zeta_function).

The transfer operator of the Gauss map <math>h(x)=1/x-\lfloor 1/x \rfloor</math> is called the [Gauss&ndash;Kuzmin&ndash;Wirsing (GKW) operator](/source/Gauss%26ndash%3BKuzmin%26ndash%3BWirsing_operator). The theory of the GKW dates back to a hypothesis by Gauss on [continued fraction](/source/continued_fraction)s and is closely related to the [Riemann zeta function](/source/Riemann_zeta_function).

==See also==
* [Bernoulli scheme](/source/Bernoulli_scheme)
* [Shift of finite type](/source/Shift_of_finite_type)
* [Krein–Rutman theorem](/source/Krein%E2%80%93Rutman_theorem)
* [Transfer-matrix method](/source/Transfer-matrix_method_(statistical_mechanics))

== References ==
* {{Cite journal
 | last = Gaspard
 | first = Pierre
 | title = r-adic one dimensional maps and the Euler summation formula
 | journal = J. Phys. A: Math. Gen.
 | volume = 25
 | pages = L483–L485
 | date = 1992 | issue = 8
 | doi = 10.1088/0305-4470/25/8/017
 | bibcode = 1992JPhA...25L.483G
 }}
* {{cite book | first=Pierre |last=Gaspard | title=Chaos, scattering and statistical mechanics | publisher=[Cambridge University Press](/source/Cambridge_University_Press) | year=1998 |isbn=0-521-39511-9 }}
* {{cite book | first=Michael C. |last=Mackey |title=Time's Arrow : The origins of thermodynamic behaviour |publisher=Springer-Verlag |year=1992 |isbn=0-387-94093-6 }}
* {{cite book | first=Dieter H. |last=Mayer | title=The Ruelle-Araki transfer operator in classical statistical mechanics | publisher=Springer-Verlag | year=1978 | isbn=0-387-09990-5}}
* {{cite book | first=David |last=Ruelle | title=Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics | publisher=Addison&ndash;Wesley, Reading | year=1978 | isbn=0-201-13504-3}}
* {{cite journal |first=David |last=Ruelle |title=Dynamical Zeta Functions and Transfer Operators |year=2002 |journal=Notices of the AMS |volume=49 |issue=8 |pages=887–895 |url=https://www.ams.org/journals/notices/200208/fea-ruelle.pdf }} ''(Provides an introductory survey).''

{{Functional analysis}}

Category:Chaos theory
Category:Dynamical systems
Category:Operator theory
Category:Spectral theory

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Adapted from the Wikipedia article [Transfer operator](https://en.wikipedia.org/wiki/Transfer_operator) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Transfer_operator?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.