On the Fibonacci–Mandelbrot set

For β∈Cβ∈C with |β|<1|β|<1 define the contractions h0(z)=βzh0(z)=βz and h1(z)=βz+1h1(z)=βz+1 and consider the attractor AβAβ of the iterated function system {h0,h1}{h0,h1}. In 1985 Barnsley and Harrington introduced the “Mandelbrot set for pairs of linear maps” which is the set of all ββ with connected attractor AβAβ. This set has been thoroughly studied by many authors. In the present paper we consider a variant of this Mandelbrot set. In particular, we consider the attractors of the iterated function system {f0,f1}{f0,f1} given by f0(z)=βz,f1(z)=1+β2z and study the associated Mandelbrot set MM. Among other things we show that MM is connected. The structure of the iterated function system {f0,f1}{f0,f1} is related to the Fibonacci Language which is the subshift of finite type over {0,1}{0,1} defined by forbidding the occurrence of two consecutive ones. This language and its difference language play an important role in our proofs.