Transfer operator

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. The transfer operator is sometimes called the Ruelle operator, after David Ruelle, or the Ruelle-Perron-Frobenius operator in reference to the applicability of the Frobenius-Perron theorem to the determination of the eigenvalues of the operator.

The iterated function to be studied is a map $f:X\rightarrow X$ for an arbitrary set $X$. The transfer operator is defined as an operator ${\mathcal {L}}$ acting on the space of functions $\Phi :X\rightarrow \mathbb {C}$ as

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

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

Some questions about the form and nature of a transfer operator are addressed in the theory of composition operators.

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 problems, such as quantum chaos and statistical mechanics, where attention is focused on the time evolution of smooth functions.

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

The eigenfunctions of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantum 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 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 $b(x)=2x-\lfloor 2x\rfloor$ is exactly solvable and is a classic example of 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 (GKW) operator and due to its extraordinary difficulty, has not been fully solved. The theory of the GKW dates back to a hypothesis by Gauss on continued fractions and is closely related to the Riemann zeta function.

See also

• Bernoulli scheme
• Shift of finite type

References

• David Ruelle (1978). Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics. Addison-Wesley, Reading. ISBN 0-201-13504-3.

• Dieter H. Mayer (1978). The Ruelle-Araki transfer operator in classical statistical mechanics. Springer-Verlag. ISBN 0-387-09990-5.

• David Ruelle, Dynamical Zeta Functions and Transfer Operators, (2002) Institut des Hautes Etudes Scientifiques preprint IHES/M/02/66. (Provides an introductory survey).