In order to understand the dynamics of the Hénon map, in 1985 Lauwerier introduced the following map: \(F_{a,b}(x,y)= (bx(1-2y)+y, ay(1-y))\), where \((x,y)\in [0,1]\times [0,1]\), \((a,b)\in [0,4]\times [0,1]\); computer simulation indicates that it possesses a complex structure similar to that of the Hénon map. The author proves that for parameter \(a\) in a set with positive measure, the corresponding Lauwerier map possesses a nontrivial topologically transitive attractor \(\Lambda\), which is the closure of the unstable set of some fixed point. Periodic points are dense in \(\Lambda\) and all of them are hyperbolic. The author also constructs the SBR measure supported on the attractor and studies its properties.