The Frobenius Perron Operator $P: L^1 \to L^1$ is defined by the integral equation $$ \int_A Pf(x) \mu(dx) = \int_{S^{-1}(A)} f(x) \mu(dx)$$ for some $\mu$-non singular map $S$. I found it in the book from 1985 called 'Probabilistic Properties of Deterministic Systems'. But I am not sure who originally invented it. Is there any history known about this subject?


The earliest occurrence of this name in Mathscinet database is the paper MR0486428 Walters, Peter A generalized Ruelle Perron-Frobenius theorem and some applications. International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), pp. 183–192. Asterisque, No. 40, Soc. Math. France, Paris, 1976.

The operator itself was introduced by D. Ruelle (Comm. Math. Phys. 9 (1968), 267–278). I am not sure whether Ruelle himself used this name for it. Some people call it the Ruelle-Perron-Frobenius operator.

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    $\begingroup$ I have read the article from D. Ruelle from 1968 but I could not find that he introduces the PF operator there. I also found in the book 'A collection of mathematical problems' from Ulam in 1960 on page 73 that he was talking about the operator, but he called it Frobenius-Perron operator. He cites O.W. Rechard from the article MR0079632 from 1956 but I do not have free acess to this paper, so I dont know if he took it again from someone else. However, in 1968 this operator was at least 12 years old. $\endgroup$ – Adam Jan 14 '15 at 12:49

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