Large deviation for weak Gibbs measures and multifractal spectra

By Marc Kessebohmer. 2001

49 citations recorded by the source corpus.

We introduce the class of `medium varying functions' and corresponding weak Gibbs measures both defined on a symbolic shift space. We prove that the Helmholtz free energy of the stochastic process of a randomly stopped Birkhoff sum measured by a weak Gibbs measure can be expressed in terms of the topological pressure. This leads to the notion of the multifractal entropy function which provides large deviation bounds. We show that the multifractal entropy functions coincide (up to constants) with multifractal spectra when Gibbs or g-measures are involved.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8850f03dd0dbd69bd579b864debdba2aeb06af61

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

1. Wen Huang
2. De-Jun Feng
3. Bernd Stratmann
4. J. Barral
5. M. Mensi
6. Marc Kessebohmer
7. Stéphane Seuret
8. Huyi Hu

In Mathematical Physics

1. Wen Huang
2. De-Jun Feng
3. Bernd Stratmann
4. Marc Kessebohmer
5. J. Barral
6. M. Mensi
7. Xiangdong Ye
8. A. Kaenmaki

In Statistical and Nonlinear Physics

1. Wen Huang
2. De-Jun Feng
3. J. Barral
4. M. Mensi
5. Bernd Stratmann
6. Marc Kessebohmer
7. A. Kaenmaki
8. Huyi Hu

Citations in the corpus

• Lyapunov Spectrum of Asymptotically Sub-additive Potentials (2009, background): “In this case, the topological entropy (or the Hausdorff dimension) of the level sets of Birkhoff averages has been extensively studied in the recent two decades (see, e.g., [13, 7, 41, 9, 20, 21, 34, 27, 8, 42, 47, 23, 16, 22, 1, 2, 29] and references therein).”
• A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates (2005, background): “Therefore, the results of [11] are applicable, and hence in this way one could immediately obtain some Large Deviation results for the situation here.”
• Gibbs measures on self-affine Sierpiński carpets and their singularity spectrum (2007, methodology): “This function has been studied extensively for measures obtained as geometric realizations of Gibbs measures defined on a symbolic space [1, 4, 5, 9, 12, 14, 22, 24, 25, 27].”