Piecewise convex transformations with no finite invariant measure

Tomasz Komorowski

Annales Polonici Mathematici (1991)

  • Volume: 54, Issue: 1, page 59-68
  • ISSN: 0066-2216

Abstract

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 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].

How to cite

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Tomasz Komorowski. "Piecewise convex transformations with no finite invariant measure." Annales Polonici Mathematici 54.1 (1991): 59-68. <http://eudml.org/doc/266163>.

@article{TomaszKomorowski1991,
abstract = { 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].},
author = {Tomasz Komorowski},
journal = {Annales Polonici Mathematici},
keywords = {piecewise convex transformation; finite invariant absolutely continuous measure; Thaler's rule-of-thumb},
language = {eng},
number = {1},
pages = {59-68},
title = {Piecewise convex transformations with no finite invariant measure},
url = {http://eudml.org/doc/266163},
volume = {54},
year = {1991},
}

TY - JOUR
AU - Tomasz Komorowski
TI - Piecewise convex transformations with no finite invariant measure
JO - Annales Polonici Mathematici
PY - 1991
VL - 54
IS - 1
SP - 59
EP - 68
AB -  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].
LA - eng
KW - piecewise convex transformation; finite invariant absolutely continuous measure; Thaler's rule-of-thumb
UR - http://eudml.org/doc/266163
ER -

References

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  1. [1] R. Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phya. 69 (1979), 1-17. Zbl0421.28016
  2. [2] S. R. Foguel, The Ergodic Theory of Markov Processes, van Nostrand Reinhold, New York 1969. 
  3. [3] P. Kacprowski, On the existence of invariant measures for piecewise smooth transformations, Ann. Polon. Math. 40 (1983), 179-184. 
  4. [4] A. Lasota and M. C. Mackey, Probabilistic Properties of Deterministic Systems, Cambridge University Press, 1985. Zbl0606.58002
  5. [5] A. Lasota and J. A. Yorke, Exact dynamical systems and the Frobenius- Perron operators, Trans. Amer. Math. Soc. 273 (1982), 375-384. Zbl0524.28021
  6. [6] F. Schweiger, Numbertheoretical endomorphisms with a-finite invariant measure, Israel J. Math. 21 (1975), 308-318. Zbl0314.10037
  7. [7] F. Schweiger, Some remarks on ergodicity and invariant measures, Michigan Math. J. 22 (1975), 308-318. 
  8. [8] M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math. 37 (1980), 303-314. Zbl0447.28016
  9. [9] M. Thaler, Transformations on [0, 1] with infinite invariant measure, ibid. 46 (1983), 67-96. 

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