# American Institute of Mathematical Sciences

June  2012, 32(6): 1977-1995. doi: 10.3934/dcds.2012.32.1977

## On the higher-dimensional multifractal analysis

 1 LAGA (UMR 7539), Département de Mathématiques, Institut Galilée, Université Paris 13, Villetaneuse 2 Department of Mathematics, Tsinghua University, Beijing

Received  April 2011 Revised  September 2011 Published  February 2012

We achieve the higher-dimensional multifractal analysis for quotients of almost additive potentials on topologically mixing subshifts of finite type without restriction on the regularity of the potentials, nor on the support of the Hausdorff spectrum, for which we do not need to assume that it has a non empty interior.
Citation: Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977
##### References:
 [1] J. Barral and Y. H. Qu, Localized asymptotic behavior for almost additive potentials, to appear in Discrete Contin. Dyn. Syst.,, \arXiv{1104.1442v1}., ().   Google Scholar [2] A. de Acosta, A general non-convex large deviation result with applications to stochastic equations,, Probab. Theory Related Fields, 118 (2000), 483.  doi: 10.1007/PL00008752.  Google Scholar [3] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. & Dynam. Sys., 16 (1996), 871.   Google Scholar [4] L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures,, Discrete Contin. Dyn. Syst., 16 (2006), 279.  doi: 10.3934/dcds.2006.16.279.  Google Scholar [5] L. Barreira and P. Doutor, Almost additive multifractal analysis,, J. Math. Pures Appl. (9), 92 (2009), 1.  doi: 10.1016/j.matpur.2009.04.006.  Google Scholar [6] L. Barreira, B. Saussol and J. Schmeling, Higher-dimensional multifractal analysis,, J. Math. Pures Appl. (9), 81 (2002), 67.  doi: 10.1016/S0021-7824(01)01228-4.  Google Scholar [7] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes in Mathematics, 470 (1975).   Google Scholar [8] H. Cajar, "Billingsley Dimension in Probability Spaces,", Lecture Notes in Mathemaitcs, 892 (1981).   Google Scholar [9] Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials,, Discrete Contin. Dyn. Syst., 20 (2008), 639.   Google Scholar [10] K. J. Falconer, A subadditive thermodynamic formalism for mixing repellers,, J. Phys. A, 21 (1988).  doi: 10.1088/0305-4470/21/14/005.  Google Scholar [11] A.-H. Fan and D.-J. Feng, On the distribution of long-term time averages on symbolic space,, J. Statist. Phys., 99 (2000), 813.  doi: 10.1023/A:1018643512559.  Google Scholar [12] A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy,, J. London Math. Soc. (2), 64 (2001), 229.  doi: 10.1017/S0024610701002137.  Google Scholar [13] D.-J. Feng, The variational principle for products of non-negative matrices,, Nonlinearity, 17 (2004), 447.  doi: 10.1088/0951-7715/17/2/004.  Google Scholar [14] D.-J. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials,, Commun. Math. Phys., 297 (2010), 1.  doi: 10.1007/s00220-010-1031-x.  Google Scholar [15] D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices,, Math. Res. Lett., 9 (2002), 363.   Google Scholar [16] D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers,, Adv. Math., 169 (2002), 58.  doi: 10.1006/aima.2001.2054.  Google Scholar [17] D.-J. Feng and E. Olivier, Multifractal analysis of the weak Gibbs measures and phase transition-Application to some Bernoulli convolutions,, Ergod. Th. & Dynam. Sys., 23 (2003), 1751.   Google Scholar [18] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps,, Ergod. Th. & Dynam. Sys., 17 (1997), 147.   Google Scholar [19] A. Mummert, The thermodynamic formalism for almost-additive sequences,, Discrete Contin. Dyn. Syst., 16 (2006), 435.  doi: 10.3934/dcds.2006.16.435.  Google Scholar [20] Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions,, J. Statist. Phys., 86 (1997), 233.  doi: 10.1007/BF02180206.  Google Scholar [21] Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, Mathematical Foundation, and Examples,, Chaos, 7 (1997), 89.  doi: 10.1063/1.166242.  Google Scholar [22] D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters,, Ergod. Th. & Dynam. Sys., 9 (1989), 527.   Google Scholar [23] R. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar [24] D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Encyclopedia of Mathematics and its Applications, 5 (1978).   Google Scholar [25] F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets,, Ergod. Th. & Dynam. Sys., 23 (2003), 317.   Google Scholar

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##### References:
 [1] J. Barral and Y. H. Qu, Localized asymptotic behavior for almost additive potentials, to appear in Discrete Contin. Dyn. Syst.,, \arXiv{1104.1442v1}., ().   Google Scholar [2] A. de Acosta, A general non-convex large deviation result with applications to stochastic equations,, Probab. Theory Related Fields, 118 (2000), 483.  doi: 10.1007/PL00008752.  Google Scholar [3] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. & Dynam. Sys., 16 (1996), 871.   Google Scholar [4] L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures,, Discrete Contin. Dyn. Syst., 16 (2006), 279.  doi: 10.3934/dcds.2006.16.279.  Google Scholar [5] L. Barreira and P. Doutor, Almost additive multifractal analysis,, J. Math. Pures Appl. (9), 92 (2009), 1.  doi: 10.1016/j.matpur.2009.04.006.  Google Scholar [6] L. Barreira, B. Saussol and J. Schmeling, Higher-dimensional multifractal analysis,, J. Math. Pures Appl. (9), 81 (2002), 67.  doi: 10.1016/S0021-7824(01)01228-4.  Google Scholar [7] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes in Mathematics, 470 (1975).   Google Scholar [8] H. Cajar, "Billingsley Dimension in Probability Spaces,", Lecture Notes in Mathemaitcs, 892 (1981).   Google Scholar [9] Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials,, Discrete Contin. Dyn. Syst., 20 (2008), 639.   Google Scholar [10] K. J. Falconer, A subadditive thermodynamic formalism for mixing repellers,, J. Phys. A, 21 (1988).  doi: 10.1088/0305-4470/21/14/005.  Google Scholar [11] A.-H. Fan and D.-J. Feng, On the distribution of long-term time averages on symbolic space,, J. Statist. Phys., 99 (2000), 813.  doi: 10.1023/A:1018643512559.  Google Scholar [12] A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy,, J. London Math. Soc. (2), 64 (2001), 229.  doi: 10.1017/S0024610701002137.  Google Scholar [13] D.-J. Feng, The variational principle for products of non-negative matrices,, Nonlinearity, 17 (2004), 447.  doi: 10.1088/0951-7715/17/2/004.  Google Scholar [14] D.-J. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials,, Commun. Math. Phys., 297 (2010), 1.  doi: 10.1007/s00220-010-1031-x.  Google Scholar [15] D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices,, Math. Res. Lett., 9 (2002), 363.   Google Scholar [16] D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers,, Adv. Math., 169 (2002), 58.  doi: 10.1006/aima.2001.2054.  Google Scholar [17] D.-J. Feng and E. Olivier, Multifractal analysis of the weak Gibbs measures and phase transition-Application to some Bernoulli convolutions,, Ergod. Th. & Dynam. Sys., 23 (2003), 1751.   Google Scholar [18] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps,, Ergod. Th. & Dynam. Sys., 17 (1997), 147.   Google Scholar [19] A. Mummert, The thermodynamic formalism for almost-additive sequences,, Discrete Contin. Dyn. Syst., 16 (2006), 435.  doi: 10.3934/dcds.2006.16.435.  Google Scholar [20] Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions,, J. Statist. Phys., 86 (1997), 233.  doi: 10.1007/BF02180206.  Google Scholar [21] Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, Mathematical Foundation, and Examples,, Chaos, 7 (1997), 89.  doi: 10.1063/1.166242.  Google Scholar [22] D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters,, Ergod. Th. & Dynam. Sys., 9 (1989), 527.   Google Scholar [23] R. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar [24] D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Encyclopedia of Mathematics and its Applications, 5 (1978).   Google Scholar [25] F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets,, Ergod. Th. & Dynam. Sys., 23 (2003), 317.   Google Scholar
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