Summary
Let\(P_\varphi :L^1 (R) \to L^1 (R)\) be the Frobenius-Perron operator corresponding to a nonsingular point-transformationϕ of the real lineR into itself and let for each natural numbern, P n be the discrete analogue ofP fϕ. It is shown that under fairly weak restrictions onϕ, the equationf=P fϕ f has an unique solutionf 0 such thatf 0>0 (a.e.), ¦‖f 0¦‖=1, and that this solution can be approximated inL 1 (R) in two different ways: (1) by the sequence\(\left\{ {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}\sum\limits_{k = 0}^{n - 1} {P_\varphi ^k f} } \right\}\) wheref≧0, ¦‖f¦‖=1, and (2) by the sequence {s On } of simple functions such thats n=Pn(s On ).
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Bugiel, P. Approximation for the measures of ergodic transformations of the real line. Z. Wahrscheinlichkeitstheorie verw Gebiete 59, 27–38 (1982). https://doi.org/10.1007/BF00575523
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DOI: https://doi.org/10.1007/BF00575523