On <i>q</i>-deformed logistic maps

We consider the logistic family \begin{document}$ f_{a} $\end{document} and a family of homeomorphisms \begin{document}$ \phi _{q} $\end{document} . The \begin{document}$ q $\end{document} -deformed system is given by the composition map \begin{document}$ f_{a}\circ \phi _{q} $\end{document} . We study when this system has non zero fixed points which are LAS and GAS. We also give an alternative approach to study the dynamics of the \begin{document}$ q $\end{document} -deformed system with special emphasis on the so-called Parrondo's paradox finding parameter values \begin{document}$ a $\end{document} for which \begin{document}$ f_{a} $\end{document} is simple while \begin{document}$ f_{a}\circ \phi _{q} $\end{document} is dynamically complicated. We explore the dynamics when several \begin{document}$ q $\end{document} -deformations are applied.