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arXiv · 0706.2580

Planar maps whose second iterate has a unique fixed point

Begoña Alarcón·Carlos Gutierrez·José Martínez-Alfaro
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Abstract

Let a>0, F: R^2 -> R^2 be a differentiable (not necessarily C^1) map and Spec(F) be the set of (complex) eigenvalues of the derivative F'(p) when p varies in R^2. (a) If Spec(F) is disjoint of the interval [1,1+a[, then Fix(F) has at most one element, where Fix(F) denotes the set of fixed points of F. (b) If Spec(F) is disjoint of the real line R, then Fix(F^2) has at most one element. (c) If F is a C^1 map and, for all p belonging to R^2, the derivative F'(p) is neither a homothety nor has simple real eigenvalues, then Fix(F^2) has at most one element, provided that Spec(F) is disjoint of either (c1) the union of the number 0 with the intervals ]-\infty, -1] and [1,\infty[, or (c2) the interval [-1-a, 1+a]. Conditions under which Fix(F^n), with n>1, is at most unitary are considered.

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Begoña Alarcón, Carlos Gutierrez, José Martínez-Alfaro. 2007-06-18. Planar maps whose second iterate has a unique fixed point. https://arxiv.org/abs/0706.2580

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