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 ${\displaystyle f:X\rightarrow X}$ for an arbitrary set ${\displaystyle X}$. The transfer operator is defined as an operator ${\displaystyle {\mathcal {L}}}$ acting on the space of functions ${\displaystyle \{\Phi :X\rightarrow \mathbb {C} \}}$ as

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