Abstract. The Hausdor and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, re ecting geometric properties of the limit set, are introduced, proven to exist, and to be unique. The existence of a unique invariant probability equivalent to the conformal measure is derived. Our methods employ the concepts of the Perron-Frobenius operator, symbolicdynamics on an in nite dimensional shift space, and the properties of the above mentioned ergodic invariant measure. A formula for the Hausdor dimension of the limit set in terms of the pressure function is derived. Fractal phenomena not exhibited by nite systems are shown to appear in the in nite case. In particular a variety of conditions are provided for Hausdor and packing measures to be positive or nite, and a number of examples are described showing the appearance of various possible combinations for these quantities. One example given special attention is the limit set associated to the complex continued fraction expansion { in particular lower and upper estimates for its Hausdor dimension are given. A large natural class of systems whose limit sets are "dimensionless in the restricted sense" is described.

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A parabolic rational map of the Riemann sphere admits a nonatomic /¡-conformai measure on its Julia set where h = the HausdorfT dimension of the Julia set and satisfies 1/2 < h < 2 . With respect to this measure the rational map is conservative, exact and there is an equivalent cr-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework.for every Borel set A c J(T) such that T\A is injective (see [34]).It turns out (Theorem 8.7) that, for h = the HausdorfT dimension of J(T), the unique A-conformal measure for T is nonatomic. This result is obtained