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Hyperbolicity in a class of one-dimensional maps

Publicacions Matemàtiques
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Publicacions Matemátiques, Vol 34 (1990, 93-105 . Abstract HYPERBOLICITY IN A CLASS OF ONE-DIMENSIONAL MAPS GREGORY J . DAVIs In this paper we provide a direct proof of hyperbolicity for a class of one- dimensional maps on the unit interval . The maps studied are degenerate forms of the standard quadratic map on the interval . These maps are important in understanding the Newhouse theory of infinitely many sinks due to homoclinic tangencies in two dimensions . Introduction In the theory of infinitely many sinks, two-dimensional invariant sets are formed when homoclinic tangencies between stable and unstable manifolds of a hyperbolic periodic point are formed . In order to show that infinitely many sinks occur in this situation, we must show that these invariant sets are hyper- bolic ([1], [4], or [6]), which is a major undertaking . When the homoclinic tangency is quadratic in nature, the two-dimensional problem has been thought of as a perturbation of the one-dimensional map ([3], [5]) . In [5], a more complete and elegant proof was obtained by conjucating the hyperbolic invariant set for the quadratic map fb(x) = bx(1 - x) to the two- dimensional invariant set in the two-dimensional infinitely many sinks problem . In the case where the homoclinic tangency is degenerate (Le ., of order r, r = 4,6 . . . . ), the present proof is very long and involved [1] . If a conjucacy between the one and two-dimensional degenerate problems can be determined, then it may be possible to treat the higher order tangencies in two dimensions using the same type of ideas as presented in [5] . With the above motivation, we will examine the hyperbolicity of the following family of one-dimensional maps on the unit interval . The one-dimensional maps that we are concerned with are of the form r fb(x) = b[2r - Cx - 2) 1, b>2r, r=4,6,8, . . . where b is a real parameter and r is a fixed positive even integer . By studying these one-dimensional maps we will gain insight as to how the hyperbolicity

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