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Generalized bounded variation and applications to piecewise monotonic transformations. (English) Zbl 0574.28014

Authors summary: We prove the quasi-compactness of the Perron-Frobenius operator of piecewise monotonic transformations when the inverse of the derivative is Hölder continuous or, more generally, of bounded p- variation.
Additional remark: The result has consequences for the study of the one- dimensional Poincaré-map of the Lorenz-flow as pointed out by [S. Wong: J. Lond. Math. Soc., II. Ser. 22, 506-520 (1980; Zbl 0459.28022)] and [C. Robinson, Ergodic Theory Dyn. Syst. 4, 605-611 (1984; Zbl 0558.28011)]. Robinson also attempts to give a much shorter proof of the above result (reducing it to a result of Wong). However, his proof is wrong.

MSC:

28D05 Measure-preserving transformations
37A99 Ergodic theory
47A35 Ergodic theory of linear operators
60F05 Central limit and other weak theorems
26A45 Functions of bounded variation, generalizations
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