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 for an arbitrary set . The transfer operator is defined as an operator acting on the space of functions as
where is an auxiliary valuation function. When has a Jacobian determinant, then is usually taken to be .
The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic pushforward of g: in essence, the transfer operator is the direct image functor in the category of measurable spaces. The left-adjoint of the Frobenius–Perron operator is the Koopman operator or composition operator.
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