The dynamics of \(\lambda+z+\exp(z)\) (Q1269049)

The paper is devoted to a dynamics of the map \(f_\lambda(z)=\lambda+z+e^z\) both for \(\lambda>0\) and \(\lambda=\alpha+i\beta\), \(\beta\neq 0\). First of all, the authors prove that the set of points that share the same refined itinerary and tend monotonically to infinity lie on a Lipschitz curve (the case of \(\lambda>0\)). In the case of \(\lambda=\alpha+i\beta\), \(\beta\neq 0\), they prove that for some values of the parameter there exist invariant curves which divide the complex plane into bent stripes giving rise to symbolic dynamics similar to those in the first case.