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}., ().

[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.

[3]

L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. & Dynam. Sys., 16 (1996), 871.

[4]

L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures,, Discrete Contin. Dyn. Syst., 16 (2006), 279. doi: 10.3934/dcds.2006.16.279.

[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.

[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.

[7]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes in Mathematics, 470 (1975).

[8]

H. Cajar, "Billingsley Dimension in Probability Spaces,", Lecture Notes in Mathemaitcs, 892 (1981).

[9]

Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials,, Discrete Contin. Dyn. Syst., 20 (2008), 639.

[10]

K. J. Falconer, A subadditive thermodynamic formalism for mixing repellers,, J. Phys. A, 21 (1988). doi: 10.1088/0305-4470/21/14/005.

[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.

[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.

[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.

[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.

[15]

D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices,, Math. Res. Lett., 9 (2002), 363.

[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.

[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.

[18]

D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps,, Ergod. Th. & Dynam. Sys., 17 (1997), 147.

[19]

A. Mummert, The thermodynamic formalism for almost-additive sequences,, Discrete Contin. Dyn. Syst., 16 (2006), 435. doi: 10.3934/dcds.2006.16.435.

[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.

[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.

[22]

D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters,, Ergod. Th. & Dynam. Sys., 9 (1989), 527.

[23]

R. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).

[24]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Encyclopedia of Mathematics and its Applications, 5 (1978).

[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.

show all references

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}., ().

[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.

[3]

L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. & Dynam. Sys., 16 (1996), 871.

[4]

L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures,, Discrete Contin. Dyn. Syst., 16 (2006), 279. doi: 10.3934/dcds.2006.16.279.

[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.

[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.

[7]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes in Mathematics, 470 (1975).

[8]

H. Cajar, "Billingsley Dimension in Probability Spaces,", Lecture Notes in Mathemaitcs, 892 (1981).

[9]

Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials,, Discrete Contin. Dyn. Syst., 20 (2008), 639.

[10]

K. J. Falconer, A subadditive thermodynamic formalism for mixing repellers,, J. Phys. A, 21 (1988). doi: 10.1088/0305-4470/21/14/005.

[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.

[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.

[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.

[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.

[15]

D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices,, Math. Res. Lett., 9 (2002), 363.

[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.

[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.

[18]

D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps,, Ergod. Th. & Dynam. Sys., 17 (1997), 147.

[19]

A. Mummert, The thermodynamic formalism for almost-additive sequences,, Discrete Contin. Dyn. Syst., 16 (2006), 435. doi: 10.3934/dcds.2006.16.435.

[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.

[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.

[22]

D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters,, Ergod. Th. & Dynam. Sys., 9 (1989), 527.

[23]

R. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).

[24]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Encyclopedia of Mathematics and its Applications, 5 (1978).

[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.

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