Dimensions and Measures in Infinite Iterated Function Systems

The Hausdorff 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, reflecting geometric properties of the limit set, are introduced, proved 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, symbolic dynamics on a shift space with an infinite alphabet, and the properties of the above-mentioned ergodic invariant measure. A formula for the Hausdorff dimension of the limit set in terms of the pressure function is derived. Fractal phenomena not exhibited by finite systems are shown to appear in the infinite case. In particular, a variety of conditions are provided for Hausdorff and packing measures to be positive or finite, 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 Hausdorff dimension are given. A large natural class of systems whose limit sets are ‘dimensionless in the restricted sense’ is described.