Calculation of Lefschetz and Nielsen Numbers in Hyperspaces for Fractals and Dynamical Systems

A simple argument is given as to why it is always trivial to calculate Lefschetz and Nielsen numbers for iterated function systems or dynamical systems in hyperspaces. The problem is reduced to a simple combinatorical situation on a finite set. In the papers [3], [4] the question was raised whether the Lefschetz or even the Nielsen number might be used in the hyperspace H(X) of all nonempty compact subsets of a metric space X to prove the existence of one or several fractals of a singleor multivalued iterated function system (IFS). This is also a question which arises naturally for discrete dynamical systems if studied from the viewpoint of hyperspaces [2], [17]. It is the purpose of this note to observe that it is possible to calculate these numbers and that this task actually is surprisingly trivial. In fact, we show that the situation is as simple as it can be: It can be completely understood by counting fixed points of a map of a finite set of typically very small cardinality. The reason for this is that hyperspaces are—despite of the fact that they lack any evident vector space structure—topologically as simple as one can expect. Moreover, this is not only true for the hyperspace H(X) of nonempty compact subsets (endowed with the Hausdorff metric), but actually even for each growth hyperspace. Recall that a subset G ⊆ H(X) is called a growth hyperspace of X if G is closed in H(X) and has the following property: Whenever A ∈ G and B ∈ H(X) are such that A ⊆ B and each component of B meets A, then B ∈ G. Examples. (1) G = H(X) is a growth hyperspace. (2) The family H1(X) of all connected compact nonempty subsets of X is a growth hyperspace. More generally, for each n = 1, 2, . . . , the space G = Hn(X) consisting of all nonempty compact subsets with at most n components is a growth hyperspace. Note that the space of all nonempty Received by the editors March 24, 2005 and, in revised form, September 20, 2005. 2000 Mathematics Subject Classification. Primary 37B99; Secondary 47H04, 47H09, 47H10, 54H25.