Switched Server Systems Whose Parameters are Normal Numbers in Base 4

Switched server systems are mathematical models of manufacturing, traffic and queueing systems. Recently, it was proved in (Eur. J. Appl. Math. 31(4) (2020), pp. 682-708) that there exist switched server systems with 3 buffers (tanks), a server, filling rates ρ1 = ρ2 = ρ3 = 1 3 and parameters d1, d2, d3 > 0 whose global attractor is a fractal set. In this article, we prove that if x1 in (0, 1 3 ), x2 in ( 1 3 , 2 3 ) and x3 in ( 2 3 , 1) are rational numbers or normal numbers in base 4 (or more generally, rich numbers to base 4) and (d1, d2, d3) is the vector with positive entries satisfying d1 = 1 3x1 − 1, d2 = 2− 3x2 3x2 − 1 , d3 = 3− 3x3 3x3 − 2 , then the corresponding switched server has no fractal attractor. More precisely, the Poincaré map of the system has a finite global attractor. The approach we use is to study the topological dynamics of a family of piecewise λ-affine contractions that includes the Poincaré map of the switched server system as a particular case.