Summary
We present two examples of Markov maps which satisfy the expanding condition, Rényi's condition and do not admit any absolutely continuous invariant measures. These examples are counterexamples to the theorem formulated in [1, p. 1].
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Bugiel, P.: Approximation for the measures of ergodic transformations of the real line. Z. Wahrscheinlichkeitstheor. Verw. Geb. 59, 27–38 (1982)
Lasota, A., Yorke, J.A.: On the existence of invariant measures for piecewise monotonic transformations. Trans. Am. Math. Soc. 186, 481–488 (1973)
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Bugiel, P. A note on invariant measures for Markov maps of an interval. Z. Wahrscheinlichkeitstheorie verw Gebiete 70, 345–349 (1985). https://doi.org/10.1007/BF00534867
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DOI: https://doi.org/10.1007/BF00534867