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By Marc (d-brmn) Stratmann, Bernd O. (-stan-sms, Reviewed Dieter and H. Mayer


A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups. (English summary) Ergodic Theory Dynam. Systems 24 (2004), no. 1, 141–170. In this paper the authors set up a thermodynamic formalism for a general class of functions defined over a subshift of finite type (X, σ). These functions are given by tuples (∆, A) of certain sequences of continuous real valued functions {∆n} and {An}, where the family of functions ∆n satisfies some growth properties with respect to the family of functions An. They introduce a generalized notion of pressure for such functions for which variational principles can be shown. In the second part of the paper these results are then applied to families ∆ of so called distance functions, which are positive, increasing and pointwise unbounded. They can be used to introduce a metric on X and the notion of Hausdorff dimension dim∆. The authors determine this dimension for the level sets F∆(a) = {x ∈ X: limn→ ∞ ∆n(x) n = a}. In the third part of the paper the general formalism is applied to derive a multifractal analysis of the limit sets L(G) of geometrically finite Kleinian groups G with possible parabolic fixed points of different ranks. Finally the authors derive necessary and sufficient conditions for the existence of phase transitions in shifts over infinite alphabets

Year: 2011
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