Abstract
Dealing with infinite iterated function systems we introduce and develop the ergodic theory of Hölder systems of functions similarly as in [HU] and [HMU]. In the context of conformal infinite iterated function systems we prove the volume lemma for the Hausdorff dimension of the projection onto the limit set of a shift invariant measure. This can be considered as a Billingsley type result. Our cenral goal is to demonstrate the appearance of the "singularity-absolute continuity" dichotomy for equilibrium states of Hölder systems of functions which has been observed in [PUZ,I] and [PUZ,II] (see also [DU1] and [DU2]) in the setting of rational functions of the Riemann sphere.
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Urbański, M. Hausdorff Measures versus Equilibrium States of Conformal Infinite Iterated Function Systems. Periodica Mathematica Hungarica 37, 153–205 (1998). https://doi.org/10.1023/A:1004742822761
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DOI: https://doi.org/10.1023/A:1004742822761