mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of

dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

quantum chaos Quantum chaos is a branch of physics focused on how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics ...

and

fractals In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...

. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the

invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mappin ...

of the system. The transfer operator is sometimes called the Ruelle operator, after

David Ruelle David Pierre Ruelle (; born 20 August 1935) is a Belgian and naturalized French mathematical physicist. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term ''strange attractor'', and devel ...

, or the Perron–Frobenius operator or Ruelle–Perron–Frobenius operator, in reference to the applicability of the

Perron–Frobenius theorem In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real. The corresponding eigenvector can be chosen to ha ...

to the determination of the

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s of the operator.

Definition

The iterated function to be studied is a map

f\colon X\rightarrow X

for an arbitrary set

X

. The transfer operator is defined as an operator

\mathcal

acting on the space of functions

\

as :

(\mathcal\Phi)(x) = \sum_ g(y) \Phi(y)

where

g\colon X\rightarrow\mathbb

is an auxiliary valuation function. When

f

has a Jacobian determinant

, J,

, then

g

is usually taken to be

g=1/, J,

. The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic

pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...

of ''g'': in essence, the transfer operator is the

direct image functor In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topo ...

in the category of

measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, an ...

s. The left-adjoint of the Perron–Frobenius operator is the

Koopman operator In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule C_\phi (f) = f \circ \phi where f \circ \phi denotes function composition. It is also encountered in composition of permutations in permutati ...

composition operator In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule C_\phi (f) = f \circ \phi where f \circ \phi denotes function composition. It is also encountered in composition of permutations in permutati ...

. 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 In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In time series analysis, the shift operator is called the '' lag opera ...

acting on a

shift space Shift may refer to: Art, entertainment, and media Gaming * ''Shift'' (series), a 2008 online video game series by Armor Games * '' Need for Speed: Shift'', a 2009 racing video game ** '' Shift 2: Unleashed'', its 2011 sequel Literature * ''S ...

. 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 A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel me ...

and the

Hessenberg matrix In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above ...

, both of which generate systems of

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...

via a right-shift.

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 Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

problems, such as

and

, where attention is focused on the time evolution of smooth functions. In turn, this has medical applications to

rational drug design Drug design, often referred to as rational drug design or simply rational design, is the invention, inventive process of finding new medications based on the knowledge of a biological target. The drug is most commonly an organic compound, organi ...

, through the field of

molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the Motion (physics), physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamics ( ...

. It is often the case that the transfer operator is positive, has discrete positive real-valued

s, with the largest eigenvalue being equal to one. For this reason, the transfer operator is sometimes called the Frobenius–Perron operator. The

eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...

s of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantum

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified 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 Support may refer to: Arts, entertainment, and media * Supporting character * Support (art), a solid surface upon which a painting is executed Business and finance * Support (technical analysis) * Child support * Customer support * Income Su ...

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 Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...

. The transfer operator of the

Bernoulli map The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) : T: , 1) \to deterministic chaos; the discrete eigenvalues correspond to the Bernoulli polynomials">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

h(x)=1/x-\lfloor 1/x \rfloor

is called the Gauss–Kuzmin–Wirsing operator, Gauss–Kuzmin–Wirsing (GKW) operator. The theory of the GKW dates back to a hypothesis by Gauss on

continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...

s and is closely related to the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

References

* * * * * * ''(Provides an introductory survey).'' {{Functional analysis Chaos theory Dynamical systems Operator theory Spectral theory

Definition

Applications

See also

References