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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 $V_{[0,x]}(1/|τ'|)$ 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.