We present here two approaches to the Dirichlet problem: The classical method of subharmonic functions that culminated in the works of Perron and Wiener, and the more modern Sobolev space approach tied to calculus of variations.

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 study algebraic properties of real positive algebraic numbers which are not less than the moduli of their conjugates. In particular, we are interested in the relation of these numbers to Perron numbers.

Let A be an n× n essentially nonnegative matrix and consider the linear differential system ẋ(t) = Ax(t), t ≥ 0. We show that there exists a constant h(A) > 0 such that the trajectory emanating from xo reaches R + at a finite time to = t(xo) ≥ 0 if and only if the sequence of points generated by a finite differences approximation from xo, with time–step 0 < h < h(A), reaches R + at a finite ind...

Concentration and equi-distribution, near the unit circle, in Solomyak’s set, of the union of the Galois conjugates and the beta-conjugates of a Parry number β are characterized by means of the Erdős-Turán approach, and its improvements by Mignotte and Amoroso, applied to the analytical function fβ(z) = −1 + ∑ i≥1 tiz i associated with the Rényi β-expansion dβ(1) = 0.t1t2 . . . of unity. Mignot...