Multifractal Analysis of the Asympyotically Additive Potentials

Lan  Xu; Long  Yang

doi:10.4236/jamp.2014.213133

Lan Xu¹^*, Long Yang²

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.

Keywords: Multifractal Analysis, u-Dimension Spectra, Asymptotically Additive

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.

References

[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.
http://dx.doi.org/10.1063/1.166232

[2] Pesin, Y. (1997) Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago Lectures in Mathematics, Chicago University Press, Chicago.
http://dx.doi.org/10.7208/chicago/9780226662237.001.0001

[3] Barreira, L. and Saussol, B. (2001) Variational Principles and Mixed Multifractal Spectra. Transactions of the American Mathematical Societyy, 353, 3919-3944.
http://dx.doi.org/10.1090/S0002-9947-01-02844-6

[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.
http://dx.doi.org/10.1007/BF02180206

[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.
http://dx.doi.org/10.1023/A:1004591209134

[6] Barreira, L., Saussol, B. and Schmeling, J. (2002) Higher-Dimensional Multifractal Analysis. Journal de Mathematiques Pures et Appliquees, 81, 67-91.
http://dx.doi.org/10.1016/S0021-7824(01)01228-4

[7] Barreira, L. and Doutor, P. (2009) Almost Additive Multifractal Analysis. Journal de Mathematiques Pures et Appliquees, 92, 1-17.
http://dx.doi.org/10.1016/j.matpur.2009.04.006

[8] Climenhaga, V. (2013) Topological Pressure of Simultaneous Level. Nonlinearity, 26, 241-268.
http://dx.doi.org/10.1088/0951-7715/26/1/241

[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.
http://dx.doi.org/10.1017/S0143385700010117

[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.
http://dx.doi.org/10.1007/BF02773211

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