Lambda-topology versus pointwise topology

Abstract

This paper deals with families of conformal iterated function systems (CIFSs). The space CIFS(X,I) of all CIFSs, with common seed space X and alphabet I, is successively endowed with the topology of pointwise convergence and the so-calledλ-topology. We show just how bad the topology of pointwise convergence is: although the Hausdorff dimension function is continuous on a dense G_δ-set, it is also discontinuous on a dense subset of CIFS(X,I). Moreover, all of the different types of systems (irregular, critically regular, etc.), have empty interior, have the whole space as boundary, and thus are dense in CIFS(X,I), which goes against intuition and conception of a natural topology on CIFS(X,I). We then prove how good the λ-topology is: Roy and Urbański [Regularity properties of Hausdorff dimension in infinite conformal IFSs. Ergod. Th. & Dynam. Sys.25(6) (2005), 1961–1983] have previously pointed out that the Hausdorff dimension function is then continuous everywhere on CIFS(X,I). We go further in this paper. We show that (almost) all of the different types of systems have natural topological properties. We also show that, despite not being metrizable (as it does not satisfy the first axiom of countability), the λ-topology makes the space CIFS(X,I) normal. Moreover, this space has no isolated points. We further prove that the conformal Gibbs measures and invariant Gibbs measures depend continuously on Φ∈CIFS(X,I) and on the parameter t of the potential and pressure functions. However, we demonstrate that the coding map and the closure of the limit set are discontinuous on an important subset of CIFS(X,I).