A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups

By Bernd Stratmann, Marc Kessebohmer. Ergodic Theory and Dynamical Systems, 2004

53 citations recorded by the source corpus.

We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these groups our revised formalisms give access to a description of the spectrum of ‘homological growth rates’ in terms of Hausdorff dimension. Furthermore, we derive necessary and sufficient conditions for the existence of ‘thermodynamic phase transitions’.

Subfields: Condensed Matter Physics, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/6fcb782f141773640672aabb12e5713ca8691406

How researchers see this paper

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In Condensed Matter Physics

1. Bernd Stratmann
2. Marc Kessebohmer
3. J. Barral
4. M. Mensi

In Mathematical Physics

1. Bernd Stratmann
2. Marc Kessebohmer
3. J. Barral
4. M. Mensi

In Molecular Biology

1. Marc Kessebohmer
2. Bernd Stratmann

Citations in the corpus

• Measures and dimensions in conformal dynamics (2003, unknown): “These have some counterparts in the theory of Kleinian groups in some particular cases (see [Bo1], [BS], and the recent article [KS] by M.”
• A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates (2005, methodology): “Secondly, note that the analysis for limit sets of Kleinian g roups in [12] did not make use of the group structure of the Kleinian group (we remark th at t e recent paper [5] gives a multifractal analysis of weak Gibbs measure, and the resul ts there are closely related to some of the results in [1”
• Fractal analysis for sets of non-differentiability of Minkowski's question mark function (2007, background): “In [19] (see also [17] [18]), we considered the n-th Stern-Brocot quotient ln and the Stern-Brocot growth rate l , which are given by (assuming that the limit exists)”