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Transactions of the American Mathematical Society

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Infinitely many co-existing sinks from degenerate homoclinic tangencies

Author: Gregory J. Davis
Journal: Trans. Amer. Math. Soc. 323 (1991), 727-748
MSC: Primary 58F15
MathSciNet review: 982238
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Abstract: The evolution of a horseshoe is an interesting and important phenomenon in Dynamical Systems as it represents a change from a nonchaotic state to a state of chaos. As we are interested in determining how this transition takes place, we are studying certain families of diffeomorphisms. We restrict our attention to certain one-parameter families $ \{ {F_t}\} $ of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points, $ {P_t};\;F_t^n({P_t}) = {P_t}$, and $ \vert\det DF_t^n({P_t})\vert < 1$. We also require the stable and unstable manifolds of $ {P_t}$ to form homoclinic tangencies as the parameter $ t$ varies through $ {t_0}$. Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values $ t$ near $ {t_0}$. We show that there are parameter values $ t$ near $ {t_0}$ at which $ {F_t}$ has infinitely many co-existing periodic sinks.

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