Editing Transfer operator

{{distinguish|transfer homomorphism}}

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

The transfer operator is sometimes called the '''Ruelle operator''', after [[David Ruelle]], or the '''Perron–Frobenius operator''' or '''Ruelle&ndash;Perron&ndash;Frobenius operator''', in reference to the applicability of the [[Perron–Frobenius theorem]] to the determination of the [[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 matrix and determinant|Jacobian]] 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 measure|pushforward]] of ''g'': in essence, the transfer operator is the [[direct image functor]] in the category of [[measurable space]]s.  The left-adjoint of the Perron&ndash;Frobenius operator is the [[Koopman operator]] or [[composition operator]]. The general setting is provided by the [[Borel functional calculus]].

As a general rule, the transfer operator can usually be interpreted as a (left-)[[shift operator]] acting on a [[shift space]]. The most commonly studied shifts are the [[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]] and the [[Hessenberg matrix]], both of which generate systems of [[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 [[Chaos theory|point dynamics]]), the transfer operator defines how (smooth) maps evolve under iteration.  Thus, transfer operators typically appear in  [[physics]] problems, such as [[quantum chaos]] and [[statistical mechanics]], where attention is focused on the time evolution of smooth functions. In turn, this has medical applications to [[rational drug design]], through the field of [[molecular dynamics]].

It is often the case that the transfer operator is positive, has discrete positive real-valued [[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]]s of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantum [[Hamiltonian (quantum theory)|Hamiltonian]], the eigenvalues will typically be very closely spaced, and thus even a very narrow and carefully selected [[quantum ensemble|ensemble]] of quantum states will encompass a large number of very different fractal eigenstates with non-zero [[support (mathematics)|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 of [[entropy]].

The transfer operator of the [[Bernoulli map]] <math>b(x)=2x-\lfloor 2x\rfloor</math> is exactly solvable and is a classic example of [[chaos theory|deterministic chaos]]; the discrete eigenvalues correspond to the [[Bernoulli polynomials]].  This operator also has a continuous spectrum consisting of the [[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 operator|Gauss&ndash;Kuzmin&ndash;Wirsing (GKW) operator]]. The theory of the GKW dates back to a hypothesis by Gauss on [[continued fraction]]s and is closely related to the [[Riemann zeta function]].

==See also==
* [[Bernoulli scheme]]
* [[Shift of finite type]]
* [[Krein–Rutman theorem]]
* [[Transfer-matrix method (statistical mechanics)|Transfer-matrix method]]

== 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]] | 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]]