Ergodic transformations from an interval into itself

Li, Tien Yien; Yorke, James A.

doi:10.1090/S0002-9947-1978-0457679-0

Ergodic transformations from an interval into itself
HTML articles powered by AMS MathViewer

by Tien Yien Li and James A. Yorke PDF

Trans. Amer. Math. Soc. 235 (1978), 183-192 Request permission

Abstract:

A class of piecewise continuous, piecewise ${C^1}$ transformations on the interval $J \subset R$ with finitely many discontinuities n are shown to have at most n invariant measures.

References

A. O. Gel′fond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 809–814 (Russian). MR 0109817
A. Lasota, Invariant measures and functional equations, Aequationes Math. 9 (1973), 193–200. MR 328026, DOI 10.1007/BF01832626
A. Lasota and James A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488 (1974). MR 335758, DOI 10.1090/S0002-9947-1973-0335758-1
T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, DOI 10.2307/2318254
W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416 (English, with Russian summary). MR 142719, DOI 10.1007/BF02020954
A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI 10.1007/BF02020331
S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. MR 0120127
A. A. Kosjakin and E. A. Sandler, Ergodic properties of a certain class of piecewise smooth transformations of a segment, Izv. Vysš. Učebn. Zaved. Matematika 3(118) (1972), 32–40 (Russian). MR 0299754