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Characterizing chaotic processes that generate uniform invariant density. (English) Zbl 1101.37024

The author defines four types of two-segmental Lebesgue processes, that is, the two-segmental processes that generate covariant density. The author proves a universal formulation of two-segmental Lebesgue processes. Geometrical characterization of two-segmental Lebesgue processes is illustrated. It is verified that for any two Lebesgue processes that belong to the same type and have an identical turning point, their horizontally linear combinations belong to the same type of Lebesgue processes.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37E05 Dynamical systems involving maps of the interval
37A05 Dynamical aspects of measure-preserving transformations
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[1] Grossmann, S.; Thomae, S., Invariant distributions and stationary correlation functions of one-dimensional discrete process, Z naturforschung, 32a, 1353-1363, (1977)
[2] CsordásAndrás, G.; Gyögyi, G.; Szépfalusy, P.; Tamás, T., Statistical properties of chaos demonstrated in a class of one-dimensional transformations, Chaos, 3, 1, 31-49, (1993)
[3] Gyögyi, G.; Szépfalusy, P., Fully developed chaotic 1-d transformations, Z phys B-conden matt, 55, 179-186, (1984)
[4] Pingel, D.; Schmelcher, P.; Diakonos, F.K., Theory and examples of the inverse Frobenius-Perron problem for complete chaotic transformations, Chaos, 9, 2, 357-366, (1999) · Zbl 0982.37024
[5] Huang, W., On complete chaotic maps with tent-map-like structures, Chaos, solitons & fractals, 24, 1, 287-299, (2005) · Zbl 1064.37029
[6] Boyarsky, A.; Góra, P., Laws of chaos: invariant measures and transformations in one dimension, (1994), Addison-Wesley Reading, MA
[7] Lasota, A.; Mackey, M.C., Probabilistic properties of deterministic transformations, (1985), Cambridge University Press Cambridge · Zbl 0606.58002
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