Abstract

For beta is an element of C with vertical bar beta vertical bar 1 define the contractions h(0)(z) = beta(z) and h(1)(z) = beta z + 1 and consider the attractor A(beta) of the iterated function system {h(0), h(1)}. In 1985 Barnsley and Harrington introduced the "Mandelbrot set for pairs of linear maps" which is the set of all beta with connected attractor A(beta). 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 {f(0), f(1)} given by f(0)(z) = beta z, f(1)(z) = 1 + beta(2)z and study the associated Mandelbrot set M. Among other things we show that M is connected. The structure of the iterated function system {f(0), f(1)} is related to the Fibonacci Language which is the subshift of finite type over {0, 1} defined by forbidding the occurrence of two consecutive ones. This language and its difference language play an important role in our proofs. (C) 2014 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.