The Perron-Frobenius theorem for homogeneous, monotone functions

  title={The Perron-Frobenius theorem for homogeneous, monotone functions},
  author={St{\'e}phane Gaubert and Jeremy Gunawardena},
  journal={Transactions of the American Mathematical Society},
If A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R + ) n . We associate a directed graph to any homogeneous, monotone function, f: (R + ) n → (R + ) n , and show that if the graph is strongly connected, then f has a (nonlinear) eigenvector in (R + ) n . Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is really about… 

Positivity Stochastic nonlinear Perron – Frobenius theorem

We establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a

A unified approach to nonlinear Perron-Frobenius theory

Let f : R >0 → R >0 be an order-preserving and homogeneous function. We show that the set of eigenvectors of f in R >0 is nonempty and bounded in Hilbert’s projective metric if and only if f

The Perron-Frobenius theorem for multi-homogeneous maps

We introduce the notion of order-preserving multi-homogeneous mapping which allows to study Perron-Frobenius type theorems and nonnegative tensors in unified fashion. We prove a weak and strong

Spectral theorem for convex monotone homogeneous maps, and ergodic control

Convergence of iterates in nonlinear Perron-Frobenius theory

  • Brian Lins
  • Mathematics
    Discrete and Continuous Dynamical Systems - B
  • 2022
. Let C be a closed cone with nonempty interior C ◦ in a Banach space. Let f : C ◦ → C ◦ be an order-preserving subhomogeneous function with a fixed point in C ◦ . We introduce a condition which

The Perron-Frobenius Theorem for Multihomogeneous Mappings

A remarkable extension of the nonlinear Perron-Frobenius theory to the multi-dimensional case is provided and poses the basis for several improvements and a deeper understanding of the current spectral theory for nonnegative tensors.

A max version of Perron--Frobenius theorem for nonnegative tensor

. In this paper we generalize the max algebra system of nonnegative matrices to the class of nonnegative tensors and derive its fundamental properties. If A ∈ (cid:60) [ m,n ] + is a nonnegative



Spectral theorem for convex monotone homogeneous maps, and ergodic control

The Contraction Mapping Approach to the Perron-Frobenius Theory: Why Hilbert's Metric?

The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0. There are many

Generalized eigenvectors and sets of nonnegative matrices

On the existence of cycle times for some non-expansive maps

We consider functions F : R n ! R n which are homogeneous and nonexpansive in thè 1 norm. We refer to these as topical functions. We study the existence of the cycle time vector (F) = lim k!1 F k (~

Invariant Half-Lines of Nonexpansive Piecewise-Linear Transformations

It is shown that if f is a nonexpansive piecewise-linear mapping of Rm into itself, there exists a unique half-line that f maps into itself and such that restriction of f thereto is a translation.

Extension of order-preserving maps on a cone

We examine the problem of extending, in a natural way, order-preserving maps that are defined on the interior of a closed cone K1 (taking values in another closed cone K2) to the whole of K1. We give


nonexpansive maps, fixed points, cycle time Dynamical systems of monotone homogeneous functions appear in Markov decision theory, in discrete event systems and in Perron-Frobenius theory. We consider