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Acta of Turin Polytechnic University in Tashkent

Abstract

It is well known that the Feigenbaum's map ϕ plays main role in theory of universality. The map ϕ is unimodal, even, analitic map of the interval [-1; 1] with one critical point. It is important that the Feigenbaum's map ϕ have infinitely many unstable periodic points and an attractor K of "Cantor type". In present work we investigate the behaviour of entrance times to the set F:

First Page

8

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11

References

[1] Feigenbaum, M.J.: Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19, 25-52 (1978), Universal metric properties of non-linear transformations. J. Stat. Phys. 21, 669-706 (1979) [2] Ya. Sinai:. Topics in Ergodic theory. Princeton University Press, Princeton, New Jersey, 1994 [3] P. Collet, J.-P. Eckmann: Iterated maps on the interval as dynamical systems, Basel, Boston, Stuttgart, Birkhauser, 1980 [4] M. Companino, H. Epstein. On the existence of Feigenbaum fixed-point. Comm. Math. Phys. 79:2, p. 261-302, 1981 [5] H. Epstein. New Proofs of the Existence of the Feigenbaum Functions Commun. Math. Phys. 106, p.395-426,1986 [6] Lanford, O.E. III: A computer-assisted proof of the Feigenbaum conjectures. Bull. Am. Math. Soc, New Series 6, p.127, 1984 [7] Lanford, O.E. III: A shorter proof of the existence of the Feigenbaum fixed point. Commun. Math. Phys. 96, p.521-538, 1984

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