JAMP  Vol.2 No.13 , December 2014
Multifractal Analysis of the Asympyotically Additive Potentials
Abstract: Multifractal analysis studies level sets of asymptotically defined quantities in dynamical systems. In this paper, we consider the u-dimension spectra on such level sets and establish a conditional variational principle for general asymptotically additive potentials by requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions.
Cite this paper: Xu, L. and Yang, L. (2014) Multifractal Analysis of the Asympyotically Additive Potentials. Journal of Applied Mathematics and Physics, 2, 1139-1148. doi: 10.4236/jamp.2014.213133.

[1]   Barreira, L., Pesin, Y. and Schmeling, J. (1997) On a General Concept of Multifractality: Multifractal Spectra for Dimensions, Entropies, and Lyapunov Exponents. Multifractal Rigidity. Chaos, 7, 27-38.

[2]   Pesin, Y. (1997) Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago Lectures in Mathematics, Chicago University Press, Chicago.

[3]   Barreira, L. and Saussol, B. (2001) Variational Principles and Mixed Multifractal Spectra. Transactions of the American Mathematical Societyy, 353, 3919-3944.

[4]   Pesin, Y. and Weiss, H. (1997) A Multifractal Analysis of Equilibrium Measures for Confromal Expanding Maps and Moran-Like Geometric Constructions. Journal of Statistical Physics, 86, 233-275.

[5]   Weiss, H. (1999) The Lyapunov Spectrum of Equilibrium Measures for Conformal Expanding Maps and Axiom-A Surface Diffeomorphisms. Journal of Statistical Physics, 95, 615-632.

[6]   Barreira, L., Saussol, B. and Schmeling, J. (2002) Higher-Dimensional Multifractal Analysis. Journal de Mathematiques Pures et Appliquees, 81, 67-91.

[7]   Barreira, L. and Doutor, P. (2009) Almost Additive Multifractal Analysis. Journal de Mathematiques Pures et Appliquees, 92, 1-17.

[8]   Climenhaga, V. (2013) Topological Pressure of Simultaneous Level. Nonlinearity, 26, 241-268.

[9]   Barreira, L. (1996) A Nonadditive Thermodynamic Formalism and Applications to Dimension Theory of Hyperbolic Dynamical Systems. Ergodic Theory and Dynamical Systems, 16, 871-927.

[10]   Barreira, L. and Schmeling, J. (2000) Sets of “Non-Typical” Points Have Full Topological Entropy and Full Hausdorff Dimension. Israel Journal of Mathematics, 116, 29-70.

[11]   Feng, D. and Huang, W. (2010) Lyapunov Spectrum of Asymptotically Sub-Additive Potential. Communications in Mathematical Physics, 297, 1-43.