Abstract
One-dimensional mappings “at the limit of period doubling” are studied in this paper without the use of the renormalization theory of Feigenbaum and others. The principal result is that the attracting part of the nonwandering set is a Cantor set of measure zero under the additional assumption that the map has negative Schwarzian derivative.
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References
Feigenbaum, M.: Universal behavior in nonlinear systems, Los Alamos Sci.1, 4–29 (1980)
Guckenheimer, J.: Sensitive dependence to initial conditions for one dimensional maps. Commun. Math. Phys.70, 133–160 (1979)
Singer, D.: Stable orbits and bifurcations of maps of the interval. SIAM J. Appl. Math.35, 260–267 (1978)
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Communicated by J.-P. Eckmann
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Guckenheimer, J. Limit sets ofS-unimodal maps with zero entropy. Commun.Math. Phys. 110, 655–659 (1987). https://doi.org/10.1007/BF01205554
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DOI: https://doi.org/10.1007/BF01205554