On an open problem regarding the spectral radius of the derivatives of a function and of its iterates

The main aim of this note is to investigate empirically the relationship between the spectral radius of the derivative of a function f : Rm → Rm and the spectral radius of the derivatives of its iterates, which is done by means of some numerical experiments for mappings of two and more variables. In this way we give a partial answer to an open problem raised in [Rus, I. A., Remark on a La Salle conjecture on global asymptotic stability, Fixed Point Theory, 17 (2016), No. 1, 159–172] and [Rus, I. A., A conjecture on global asymptotic stability, communicated at the Workshop ”Iterative Approximation of Fixed Points”, SYNASC2017, Timis¸oara, 21-24 September 2017] and also illustrate graphically the importance and difficulty of this problem in the general context. An open problem regarding the domains of convergence is also proposed.