Planar maps whose second iterate has a unique fixed point

TL;DR

This paper investigates conditions under which planar maps and their iterates have at most one fixed point, based on eigenvalue spectra and differentiability properties, contributing to fixed point theory in dynamical systems.

Contribution

It establishes new spectral conditions ensuring at most one fixed point for a map or its second iterate, extending fixed point results in planar dynamical systems.

Findings

If Spec(F) avoids [1,1+a[, then Fix(F) has at most one point.

If Spec(F) avoids R, then Fix(F^2) has at most one point.

Under certain differentiability and spectral conditions, Fix(F^2) is at most one point.

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.