Mapping stability : real rational maps of degree zero

In memory of our beloved Professor José Rodrigues Santos de Sousa Ramos (1948-2007), who João Cabral, one of the authors of this paper, had the honor of being his student between 2000 and 2006, we wrote this paper following the research by experimentation, using the new technologies to capture a new insight about a problem, as him so much love to do it. His passion was to create new relations between different fields of mathematics. He was a builder of bridges of knowledge, encouraging the birth of new ways to understand this science. One of the areas that Sousa Ramos researched was the iteration of maps and the description of its behavior, using the symbolic dynamics. So, in this issue of this journal, honoring his memory, we use experimental results to find some stable regions of a specific family of real rational maps, the ones that he worked with João Cabral. In this paper we describe a parameter space (a,b) to the real rational maps fa,b(x) = (x2 −a)/(x2 −b), using some tools of dynamical systems, as the study of the critical point orbit and Lyapunov exponents. We give some results regarding the stability of these family of maps when we iterate it, specially the ones connected to the order 3 of iteration. We hope that our results would help to understand better the behavior of these maps, preparing the ground to a more efficient use of the Kneading Theory on these family of maps, using symbolic dynamics.