What is nonlinear Perron-Frobenius theory? To get an impression of the contents of nonlinear Perron-Frobenius theory, it is useful to first recall the basics of classical Perron-Frobenius theory. Classical Perron-Frobenius theory concerns nonnegative matrices, their eigenvalues and corresponding eigenvectors. The fundamental theorems of this classical theory were discovered at the beginning of the twentieth century by Perron [179, 180], who investigated eigenvalues and eigenvectors of matrices with strictly positive entries, and by Frobenius [70-72], who extended Perron's results to irreducible nonnegative matrices. In the first section we discuss the theorems of Perron and Frobenius and some of their generalizations to linear maps that leave a cone in a finite-dimensional vector space invariant. The proofs of these classical results can be found in many books on matrix analysis, e.g., [15, 22, 73, 148, 202]. Nevertheless, in Appendix B we prove some of them once more using a combination of analytic, geometric, and algebraic methods. The geometric methods originate from work of Alexandroff and Hopf [8], Birkhoff [25], Kreȋn and Rutman [117], and Samelson [192] and underpin much of nonlinear Perron-Frobenius theory. Readers who are not familiar with these methods might prefer to first read Chapters 1 and 2 and Appendix B. Besides recalling the classical Perron-Frobenius theorems, we use this chapter to introduce some basic concepts and terminology that will be used throughout the exposition, and provide some motivating examples of classes of nonlinear maps to which the theory applies. We emphasize that throughout the book we will always be working in a finite-dimensional real vector space V , unless we explicitly say otherwise. 1.1 Classical Perron-Frobenius theory An n × n matrix A = (a i j) is said to be nonnegative if a i j ≥ 0 for all i and j. It is called positive if a i j > 0 for all i and j. Similarly, we call a vector x ∈ R n www.cambridge.org © in this web service Cambridge University Press Cambridge University Press 978-0-521-89881-2-Nonlinear Perron-Frobenius Theory Bas Lemmens and Roger Nussbaum Excerpt More information 2 What is nonlinear Perron-Frobenius theory? nonnegative (or positive) if all its coordinates are nonnegative (or positive). The spectrum of A is given by σ (A) = {λ ∈ C : Ax = λx for some x ∈ C n \ {0}}. Recall also that the spectral radius of A is given by r (A) = max{|λ| : λ ∈ σ (A)}, and satisfies the equality r (A) = lim k→∞ A k 1/k. Notice that the limit is independent of the choice of the matrix norm, or norm on R n 2 , as they are all equivalent; see Rudin [190]. The following result is due to Perron [180]. Theorem 1.1.2 (Perron-Frobenius) If A is a nonnegative irreducible n × n matrix, then the following assertions hold:

Abstract.-We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an n-simplex with n ≥ 2. Moreover, we determine the isometry group of the Hilbert geometry on the n-simplex for all n ≥ 2, and find that it has the collineation group as an index-two subgroup. These results confirm, for the class of polyhedral Hilbert geometries, several conjectures posed by P. de la Harpe.

Abstract. We investigate the iterative behaviour of continuous order preserving subhomogeneous maps f : K → K, where K is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of f converges to a periodic orbit and, moreover, the period of each periodic point of f is bounded bywhere N is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in R n , we show that the upper bound is asymptotically sharp.These results are an extension of work by Lemmens and Scheutzow concerning periodic orbits in the interior of the standard positive cone in R n .

Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and is closely related to Hilbert's classical cross-ratio metric. The version we discuss here was popularized by Bushell and can be traced back to the work of Garrett Birkhoff and Hans Samelson. It has found numerous applications in mathematical analysis, especially in the analysis of linear, and nonlinear, mappings on cones. Some of these applications are discussed in this chapter.Birkhoff's version of Hilbert's metric provides a different perspective on Hilbert geometries and naturally leads to infinite-dimensional generalizations. We illustrate this by showing some of its uses in the geometric analysis of Hilbert geometries.

Abstract. We present several results for the periods of periodic points of sup-norm nonexpansive maps. In particular, we show that the period of each periodic point of a sup-norm non-expansive map f : M → M, where M ⊂ R n , is at most max k 2 k n k . This upper bound is smaller than 3 n and improves the previously known bounds. Further, we consider a special class of sup-norm non-expansive maps, namely topical functions. For topical functions f : R n → R n Gunawardena and Sparrow have conjectured that the optimal upper bound for the periods of periodic points is n n/2 . We give a proof of this conjecture. To obtain the results we use combinatorial and geometric arguments. In particular, we analyse the cardinality of anti-chains in certain partially ordered sets. IntroductionIn this paper we are concerned with the dynamics of sup-norm non-expansive maps f : M → M, where M ⊂ R n . A characteristic property of the iterative behaviour of sup-norm non-expansive maps is that every bounded orbit converges to a periodic orbit. Moreover, there exists an upper bound for the possible periods of periodic points of supnorm non-expansive maps that only depends on the dimension of the ambient space. It is, therefore, an interesting problem to determine the optimal upper bound for the periods of periodic points of sup-norm non-expansive maps. This problem has been considered in [4, 14-16, 18, 25], and Nussbaum [18] has conjectured that 2 n is the optimal upper bound. At present, however, the conjecture is proved only for n ≤ 3 (see [15]).The main goal of the paper is to present two results concerning the periodic points of sup-norm non-expansive maps. As a first result, we show that the period of each periodic point of a sup-norm non-expansive map f : M → M, where M ⊂ R n , is at most max k 2 k n k . This upper bound is smaller than 3 n and is a considerable improvement of the previously best-known estimate n!2 n by Martus [16].