On the Structure of Lorenz Maps

We study the non-wandering set of $C^3$ contracting Lorenz maps $f$ with negative Schwarzian derivative. We show that if $f$ doesn’t have attracting periodic orbit, then there is a unique topological attractor. Precisely, there is a transitive compact set $\mathrm{\Lambda }$ such that ${\omega }_f(x)=\mathrm{\Lambda }$ for a residual set of points $x\in [0,1]$. We also develop in the context of Lorenz maps the classical theory of spectral decomposition constructed for Axiom A maps by Smale.