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On finite limit sets for transformations on the unit interval. (English) Zbl 0259.26003

MSC:
26-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to real functions
26A18 Iteration of real functions in one variable
54C99 Maps and general types of topological spaces defined by maps
05B99 Designs and configurations
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[1] Stein, P.R; Ulam, S.M, Non-linear transformation studies on electronic computers, Rozprawy mat., 39, 1-66, (1964) · Zbl 0143.18801
[2] Rechard, O.W, Invariant measures for many-one transformations, Duke math. J., 23, 477, (1956) · Zbl 0070.28001
[3] Julia, G, Mémoire sur l’itération des functions rationelles, J. de math. ser. 7, 4, 47-245, (1918), The relevant theorem appears on p. 129 ff. · JFM 46.0520.06
[4] Metropolis, N; Stein, M.L; Stein, P.R, Stable states of a non-linear transformation, Numer. math., 10, 1-19, (1967) · Zbl 0171.13103
[5] Gilbert, E.N; Riordan, J, Symmetry types of periodic sequences, Illinois J. math., 5, 657, (1961) · Zbl 0105.25101
[6] Schreier, J; Ulam, S, Eine bemerkung über die gruppe der topologischen abbildung der kreislinie auf sich selbst, Studia math., 5, 155-159, (1935) · Zbl 0013.05603
[7] See reference 1, p. 52. The result is due to S. Ulam and J. von Neumann.
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