In 1985, M. Barnsley and A. Harrington studied the iterated function system $f_c,g_c:\mathbb{C}\to \mathbb{C}$ given by $\{f_c:z\mapsto cz+1, g_c:z\mapsto cz-1\}$, where $c\in\mathbb{D}-\{0\}$. For a given value of the parameter $c$, there is a nonempty attractor in the dynamical plane. A natural subset to consider in parameter space is the corresponding connectedness locus, which we (suggestively) denote as $M$; it has been studied by several mathematicians, including T. Bousch, who proved that it is both connected, and locally connected. In 2002, C. Bandt proved that the complement of $M$ has at least two connected components; that is, $M$ is NOT full as a subset of $\mathbb{D}-\{0\}$. In this talk, we explore more of the topology of $M$. We compare/contrast the discussion of this parameter space to the study of the parameter space for quadratic polynomials $p_c:z\mapsto z^2+c$.
This is joint work with D. Calegari and A. Walker.