Abstra t. We onsider fun tions f : R n ! R n whi h are additively homogeneous and monotone in the produ t ordering on R n (topi al fun tions). We show that if some non-empty sub-eigenspa e of f is bounded in the Hilbert semi-norm then f has an additive eigenve tor and we give a Collatz-Wielandt hara terisation of the orresponding eigenvalue. The boundedness ondition is satis ed if a ertain dire...

We consider a lattice of weakly coupled expanding circle maps. We construct, via a cluster expansion of the Perron-Frobenius operator, an invariant measure for these innnite dimensional dynamical systems which exhibits space-time-chaos.

We consider a topological dynamical system T : Y ! Y on a metric space Y which forms a bre bundle over another dynamical system. If T is brewise expanding and exact along bres and if ' is a HH older continuous function we prove the existence of a system of conditional measures (called a family of Gibbs measures) where the Jacobian is determined by '. This theorem reduces to Ruelle's Perron-Frob...

The purpose of this paper is to present a unified Perron-Frobenius Theory for nonnegative, for real not necessarily nonnegative and for general complex matrices. The sign-real spectral radius was introduced for general real matrices. This quantity was shown to share certain properties with the Perron root of nonnegative matrices. In this paper we introduce the sign-complex spectral radius. Agai...

The recursively-constructed family of Mandelbrot matrices $M_n$ for $n=1$, $2$, $\ldots$ have nonnegative entries (indeed just $0$ and $1$, so each can be called a binary matrix) eigenvalues whose negatives $-\lambda = c$ give periodic orbits under the iteration, namely $z_k z_{k-1}^2+c$ with $z_0=0$, are thus contained in set. By Perron--Frobenius theorem, dominant real positive eigenvalue, wh...