Tangent Measure Distributions of Hyperbolic Cantor

Tangent measure distributions were introduced by Bandt 2] and Graf 8] as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by certain contractive mappings, which are not necessarily similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorr-or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models of Bedford and Fisher 5]. Tangent measure distributions were introduced by Bandt in 2] and, in the present form, by Graf in 8]. They provide a measure{theoretic tool to describe and understand the local geometry of sets and measures. To each point in the support of a measure, or in a set equipped with some natural measure, we assign a family of probability distributions on the space of measures, or, equivalently, a family of random measures, which reeects the behaviour of the measure as an observer zooms down towards this point. The concept of a tangent measure distribution is an extension of two ideas: On the one hand this is the idea of introducing tangent measures in the process of characterizing the regularity of measures by means of their local behaviour. This idea was used to very high eeect by Preiss in his fundamental paper 16]. Recently, tangent measures have been subject of intensive research, for a survey see 11]. On the other hand this is the idea of using an averaging procedure on the set of scales to deene, by means of ergodic theory, local characteristics of self-similar sets. This idea is due to Bedford and Fisher (see 4]). Closely related ideas can be found in the work of U. ZZ ahle on self-similar random measures (see 19]), which is continued in a series of joint work with Patzschke and M. ZZ ahle (see e.g. 18], 17]). Bandt joined these ideas and deened the tangent measure distributions of a measure at a point as limiting distributions of sequences of natural probability distributions on (rescaled) enlargements of the measure about this point. Roughly speaking, the weight that a tangent measure distribution assigns to a given set of (tangent) measures depends on the number of scales (in terms of the Haar measure on the multiplicative group of positive reals) for which the corresponding enlargement …