# Invariant measures and the compactness of the domain

Annales Polonici Mathematici (1998)

- Volume: 69, Issue: 1, page 13-24
- ISSN: 0066-2216

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topMarian Jabłoński, and Paweł Góra. "Invariant measures and the compactness of the domain." Annales Polonici Mathematici 69.1 (1998): 13-24. <http://eudml.org/doc/270416>.

@article{MarianJabłoński1998,

abstract = {We consider piecewise monotonic and expanding transformations τ of a real interval (not necessarily bounded) into itself with countable number of points of discontinuity of τ’ and with some conditions on the variation $V_\{[0,x]\}(1/|τ^\{\prime \}|)$ which need not be a bounded function (although it is bounded on any compact interval). We prove that such transformations have absolutely continuous invariant measures. This result generalizes all previous “bounded variation” existence theorems.},

author = {Marian Jabłoński, Paweł Góra},

journal = {Annales Polonici Mathematici},

keywords = {maps with unbounded oscillation; invariant measures},

language = {eng},

number = {1},

pages = {13-24},

title = {Invariant measures and the compactness of the domain},

url = {http://eudml.org/doc/270416},

volume = {69},

year = {1998},

}

TY - JOUR

AU - Marian Jabłoński

AU - Paweł Góra

TI - Invariant measures and the compactness of the domain

JO - Annales Polonici Mathematici

PY - 1998

VL - 69

IS - 1

SP - 13

EP - 24

AB - We consider piecewise monotonic and expanding transformations τ of a real interval (not necessarily bounded) into itself with countable number of points of discontinuity of τ’ and with some conditions on the variation $V_{[0,x]}(1/|τ^{\prime }|)$ which need not be a bounded function (although it is bounded on any compact interval). We prove that such transformations have absolutely continuous invariant measures. This result generalizes all previous “bounded variation” existence theorems.

LA - eng

KW - maps with unbounded oscillation; invariant measures

UR - http://eudml.org/doc/270416

ER -

## References

top- [Gó] P. Góra, Properties of invariant measures for piecewise expanding transformations with summable oscillation of the derivative, Ergodic Theory Dynam. Systems 14 (1994), 475-492. Zbl0822.28008
- [JGB] M. Jabłoński, P. Góra and A. Boyarsky, A general existence theorem for absolutely continuous invariant measures on bounded and unbounded intervals, Nonlinear World 2 (1995), 183-200. Zbl0895.28005
- [JL] M. Jabłoński and A. Lasota, Absolutely continuous invariant measures for transformations on the real line, Zeszyty Nauk. Uniw. Jagiell. Prace Mat. 22 (1981), 7-13. Zbl0479.28013
- [KS] A. A. Kosyakin and E. A. Sandler, Ergodic properties of a certain class of piecewise smooth transformations of a segment, Izv. Vyssh. Uchebn. Zaved. Mat. 1972, no. 3, 32-40 (in Russian).
- [LY] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. Zbl0298.28015
- [Re] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. Zbl0079.08901
- [Ry] M. R. Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), 69-80. Zbl0575.28011
- [Sch] B. Schmitt, Contributions à l'étude de systèmes dynamiques unidimensionnels en théorie ergodique, Ph.D. Thesis, University of Bourgogne, 1986.
- [Wo] S. Wong, Some metric properties of piecewise monotonic mappings of the unit interval, Trans. Amer. Math. Soc. 246 (1978), 493-500. Zbl0401.28011

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