Geodesic
Invariant measures and the compactness of the domain
Annales Polonici Mathematici, Tome 69 (1998) no. 1, pp. 13-24
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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 $V_{[0,x]}(1/|τ'|)$ 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. 
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Marian Jabłoński; Paweł Góra. Invariant measures and the compactness of the domain. Annales Polonici Mathematici, Tome 69 (1998) no. 1, pp. 13-24. doi: 10.4064/ap-69-1-13-24

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