Topological entropy of induced spaces for an iterated function system

An iterated function system (or IFS for short) induces two natural systems, one is on the hyperspace and the other one is on the probability measures space. The relationship between topological entropy on the original space and on the induced spaces are investigated. Specifically, we prove that the topological entropy of an IFS is zero if and only if the topological entropy of its induced system on the hyperspace is also zero. Moreover, if the topological entropy of the IFS is positive, then the topological entropy of both its induced hyperspace and probability measures space is infinite.