Cascade of sinks
Author:
Clark Robinson
Journal:
Trans. Amer. Math. Soc. 288 (1985), 841849
MSC:
Primary 58F12; Secondary 34D30
MathSciNet review:
776408
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Abstract: In this paper it is proved that if a oneparameter family of dissipative maps in dimension two creates a new homoclinic intersection for a fixed point when the parameter , then there is a cascade of quasisinks, i.e., there are parameter values converging to such that, for , has a quasisink with each point in having period . A quasisink for a map is a closed set such that each point in is a periodic point and is a quasiattracting set (à la Conley), i.e., is the intersection of attracting sets , where each has a neighborhood such that . Thus, the quasisinks are almost attracting sets made up entirely of points of period . Gavrilov and Silnikov, and later Newhouse, proved this result when the new homoclinic intersection is created nondegenerately. In this case the sets are single, isolated (differential) sinks. In an earlier paper we proved the degenerate case when the homoclinic intersections are of finite order tangency (or the family is real analytic), again getting a cascade of sinks, not just quasisinks.
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DOI:
http://dx.doi.org/10.1090/S00029947198507764088
PII:
S 00029947(1985)07764088
Article copyright:
© Copyright 1985
American Mathematical Society
