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Invariant measures for piecewise convex transformations of an interval. (English) Zbl 0998.37008

Authors’ summary: We investigate the existence and ergodic properties of absolutely continuous invariant measures for a class of piecewise monotone and convex self-maps of the unit interval. Our assumption entails a type of average convexity which strictly generalizes the case of individual branches being convex, as investigated by A. Lasota and J. A. Yorke [Trans. Am. Math. Soc. 273, 375-384 (1982; Zbl 0524.28021)]. Along with existence, we identify tractable conditions for the invariant measure to be unique and such that the system has exponential decay of correlations on bounded variation functions and Bernoulli natural extension. In the case when there is more than one invariant density we identify a dominant component over which the above properties also hold. Of particular note in our investigation is the lack of smoothness or uniform expansiveness assumptions on the map or its powers.

MSC:

37E05 Dynamical systems involving maps of the interval
37A05 Dynamical aspects of measure-preserving transformations
37A25 Ergodicity, mixing, rates of mixing
37A40 Nonsingular (and infinite-measure preserving) transformations

Citations:

Zbl 0524.28021
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