On the complete chaotic transformations that preserve symmetric invariant densities. (English) Zbl 1152.37325

Summary: A transformation \(f:[0,1]\rightarrow [0,1]\) is said to be complete chaotic if it is (i) ergodic with respect to the Lebesgue measure and (ii) chaotic in the probabilistic sense, that is, an absolutely continuous invariant density \(\varphi \) is preserved. The characteristics of the complete chaotic transformations that preserve symmetric invariant densities, that is, \(\varphi (x)=\varphi (1-x)\), for all \(x \in [0,1]\), are explored. It is found that such transformations are “invariant” with both horizontal and vertical mirroring operations in the sense that the transformations resulted do not only remain to be chaotic but also preserve an identical invariant density. Numerical examples and computer simulations are consistent with theoretical findings.


37E05 Dynamical systems involving maps of the interval
28D05 Measure-preserving transformations
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