# Piecewise convex transformations with no finite invariant measure

Annales Polonici Mathematici (1991)

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

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topTomasz 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

top- [1] R. Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phya. 69 (1979), 1-17. Zbl0421.28016
- [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. Zbl0606.58002
- [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] F. Schweiger, Numbertheoretical endomorphisms with a-finite invariant measure, Israel J. Math. 21 (1975), 308-318. Zbl0314.10037
- [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. Zbl0447.28016
- [9] M. Thaler, Transformations on [0, 1] with infinite invariant measure, ibid. 46 (1983), 67-96.

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