# 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 and Continuous Dynamical Systems, 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), 483-521. 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), 871-927. [4] L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Contin. Dyn. Syst., 16 (2006), 279-305. doi: 10.3934/dcds.2006.16.279. [5] L. Barreira and P. Doutor, Almost additive multifractal analysis, J. Math. Pures Appl. (9), 92 (2009), 1-17. 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), 67-91. 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, Springer-Verlag, Berlin-New York, 1975. [8] H. Cajar, "Billingsley Dimension in Probability Spaces," Lecture Notes in Mathemaitcs, 892, Springer-Verlag, Berlin, 1981. [9] Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657. [10] K. J. Falconer, A subadditive thermodynamic formalism for mixing repellers, J. Phys. A, 21 (1988), L737-L742. 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), 813-856. 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), 229-244. doi: 10.1017/S0024610701002137. [13] D.-J. Feng, The variational principle for products of non-negative matrices, Nonlinearity, 17 (2004), 447-457. 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), 1-43. 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), 363-378. [16] D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91. 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), 1751-1784. [18] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergod. Th. & Dynam. Sys., 17 (1997), 147-167. [19] A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Contin. Dyn. Syst., 16 (2006), 435-454. 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), 233-275. doi: 10.1007/BF02180206. [21] Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, Mathematical Foundation, and Examples, Chaos, 7 (1997), 89-106. doi: 10.1063/1.166242. [22] D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergod. Th. & Dynam. Sys., 9 (1989), 527-541. [23] R. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. [24] D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, 5, Addison-Wesley Publishing Co., Reading, Mass., 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), 317-348.

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), 483-521. 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), 871-927. [4] L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Contin. Dyn. Syst., 16 (2006), 279-305. doi: 10.3934/dcds.2006.16.279. [5] L. Barreira and P. Doutor, Almost additive multifractal analysis, J. Math. Pures Appl. (9), 92 (2009), 1-17. 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), 67-91. 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, Springer-Verlag, Berlin-New York, 1975. [8] H. Cajar, "Billingsley Dimension in Probability Spaces," Lecture Notes in Mathemaitcs, 892, Springer-Verlag, Berlin, 1981. [9] Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657. [10] K. J. Falconer, A subadditive thermodynamic formalism for mixing repellers, J. Phys. A, 21 (1988), L737-L742. 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), 813-856. 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), 229-244. doi: 10.1017/S0024610701002137. [13] D.-J. Feng, The variational principle for products of non-negative matrices, Nonlinearity, 17 (2004), 447-457. 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), 1-43. 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), 363-378. [16] D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91. 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), 1751-1784. [18] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergod. Th. & Dynam. Sys., 17 (1997), 147-167. [19] A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Contin. Dyn. Syst., 16 (2006), 435-454. 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), 233-275. doi: 10.1007/BF02180206. [21] Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, Mathematical Foundation, and Examples, Chaos, 7 (1997), 89-106. doi: 10.1063/1.166242. [22] D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergod. Th. & Dynam. Sys., 9 (1989), 527-541. [23] R. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. [24] D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, 5, Addison-Wesley Publishing Co., Reading, Mass., 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), 317-348.
 [1] Anna Mummert. The thermodynamic formalism for almost-additive sequences. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 435-454. doi: 10.3934/dcds.2006.16.435 [2] Imen Bhouri, Houssem Tlili. On the multifractal formalism for Bernoulli products of invertible matrices. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1129-1145. doi: 10.3934/dcds.2009.24.1129 [3] Vaughn Climenhaga. Multifractal formalism derived from thermodynamics for general dynamical systems. Electronic Research Announcements, 2010, 17: 1-11. doi: 10.3934/era.2010.17.1 [4] Yongluo Cao, De-Jun Feng, Wen Huang. The thermodynamic formalism for sub-additive potentials. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 639-657. doi: 10.3934/dcds.2008.20.639 [5] Julien Barral, Yan-Hui Qu. Localized asymptotic behavior for almost additive potentials. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 717-751. doi: 10.3934/dcds.2012.32.717 [6] Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857 [7] Zied Douzi, Bilel Selmi. On the mutual singularity of multifractal measures. Electronic Research Archive, 2020, 28 (1) : 423-432. doi: 10.3934/era.2020024 [8] Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial and Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259 [9] Balázs Bárány, Michaƚ Rams, Ruxi Shi. On the multifractal spectrum of weighted Birkhoff averages. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2461-2497. doi: 10.3934/dcds.2021199 [10] Hyun-Jung Kim. Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution. Communications on Pure and Applied Analysis, 2019, 18 (2) : 795-807. doi: 10.3934/cpaa.2019038 [11] Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure and Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004 [12] Miaohua Jiang. Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 967-983. doi: 10.3934/dcds.2015.35.967 [13] Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure and Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63 [14] Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627 [15] Lars Olsen. First return times: multifractal spectra and divergence points. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635 [16] Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234 [17] Vaughn Climenhaga. A note on two approaches to the thermodynamic formalism. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 995-1005. doi: 10.3934/dcds.2010.27.995 [18] Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 885-902. doi: 10.3934/cpaa.2020295 [19] Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004 [20] Jordan Emme. Hermodynamic formalism and k-bonacci substitutions. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3701-3719. doi: 10.3934/dcds.2017157

2020 Impact Factor: 1.392