Editing Transfer operator (section)

==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]].