Cascade of sinks

Author:
Clark Robinson

Journal:
Trans. Amer. Math. Soc. **288** (1985), 841-849

MSC:
Primary 58F12; Secondary 34D30

MathSciNet review:
776408

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Abstract: In this paper it is proved that if a one-parameter 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 quasi-sinks, i.e., there are parameter values converging to such that, for , has a quasi-sink with each point in having period . A quasi-sink for a map is a closed set such that each point in is a periodic point and is a quasi-attracting set (à la Conley), i.e., is the intersection of attracting sets , where each has a neighborhood such that . Thus, the quasi-sinks 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 quasi-sinks.

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0776408-8

Article copyright:
© Copyright 1985
American Mathematical Society