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On complete chaotic maps with tent-map-like structures. (English) Zbl 1064.37029

Summary: A unimodal map \(f : [0, 1]\to [0, 1]\) is said to be complete chaotic if it is both ergodic and chaotic in a probabilistic sense so as to preserve an absolutely continuous invariant measure. Sufficient conditions are provided to construct complete chaotic maps with the tent-map-like structures, that is, \(f(x) = 1 - |1 - 2g(x)|\), where \(g\) is a one-to-one onto map defined on \([0, 1]\). The simplicity and analytical characteristics of such chaotic maps simplify the calculations of various statistical properties of chaotic dynamics.

MSC:

37E05 Dynamical systems involving maps of the interval
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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