Piecewise convex transformations with no finite invariant measure

• Autorzy: Tomasz Komorowski
• Języki publikacji: EN

Abstrakty

EN

 Abstract. The paper concerns the problem of the existence of a finite invariant absolutely continuous measure for piecewise $C^2$-regular and convex transformations T: [0, l]→[0,1]. We show that in the case when T'(0) = 1 and T"(0) exists T does not admit such a measure. This result is complementary to the ones contained in [3] and [5].

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1991
• Tom: 54
• Numer: 1
• Strony: 59-68

Daty

wydano

1991

otrzymano

1989-12-04

poprawiono

1990-03-25

poprawiono

1990-06-28

Twórcy

autor

Tomasz Komorowski

• Institute of Mathematics, M. Curie-Skłodowska University, Pl. M. Curie-Skłodowskicj I, 20-031 Lublin, Poland

Bibliografia

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• [2] S. R. Foguel, The Ergodic Theory of Markov Processes, van Nostrand Reinhold, New York 1969.
• [3] P. Kacprowski, On the existence of invariant measures for piecewise smooth transformations, Ann. Polon. Math. 40 (1983), 179-184.
• [4] A. Lasota and M. C. Mackey, Probabilistic Properties of Deterministic Systems, Cambridge University Press, 1985.
• [5] A. Lasota and J. A. Yorke, Exact dynamical systems and the Frobenius- Perron operators, Trans. Amer. Math. Soc. 273 (1982), 375-384.
• [6] F. Schweiger, Numbertheoretical endomorphisms with a-finite invariant measure, Israel J. Math. 21 (1975), 308-318.
• [7] F. Schweiger, Some remarks on ergodicity and invariant measures, Michigan Math. J. 22 (1975), 308-318.
• [8] M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math. 37 (1980), 303-314.
• [9] M. Thaler, Transformations on [0, 1] with infinite invariant measure, ibid. 46 (1983), 67-96.