Topological Attractors of Contracting Lorenz Maps

We study the non-wandering set of contracting Lorenz maps. We show that if such a map $f$ doesn’t have any attracting periodic orbit, then there is a unique topological attractor. Precisely, there is a compact set $\mathrm{\Lambda }$ such that ${\omega }_f(x)=\mathrm{\Lambda }$ for a residual set of points $x\in [0,1]$. Furthermore, we classify the possible kinds of attractors that may occur.