# American Institute of Mathematical Sciences

March  2012, 32(3): 717-751. doi: 10.3934/dcds.2012.32.717

## Localized asymptotic behavior for almost additive potentials

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

Received  September 2010 Revised  December 2010 Published  October 2011

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of weak concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of $\phi_n(x)$ is localized, i.e. depends on the point $x$ rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form $\{x\in X: \lim_{n\to\infty} \phi_n(x)/n=\xi(x)\}$, where $\xi$ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in $\mathbb{R}^d$, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.
Citation: 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
##### References:
 [1] J. Barral and D. J. Feng, Weighted thermodynamic formalism and applications, arXiv:0909.4247v1. [2] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. Dynam. Sys., 16 (1996), 871-927. doi: 10.1017/S0143385700010117. [3] L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Contin. Dyn. Syst., 16 (2006), 279-305. doi: 10.3934/dcds.2006.16.279. [4] L. Barreira and P. Doutor, Almost additive multifractal analysis, J. Math. Pures Appl., 92 (2009), 1-17. doi: 10.1016/j.matpur.2009.04.006. [5] L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214 (2000), 339-371. doi: 10.1007/s002200000268. [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] R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. [9] R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J., 7 (1940), 312-340. doi: 10.1215/S0012-7094-40-00718-9. [10] G. Brown, G. Michon and J. Peyrière, On the multifractal analysis of measures, J. Stat. Phys., 66 (1992), 775-790. doi: 10.1007/BF01055700. [11] Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657. [12] P. Collet, J. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Statist. Phys., 47 (1987), 609-644. doi: 10.1007/BF01206149. [13] 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. [14] 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. [15] A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244. doi: 10.1017/S0024610701002137. [16] D.-J. Feng, Lyapounov exponents for products of matrices and multifractal analysis. Part I: Positive matrices, Israel J. Math., 138 (2003), 353-376. doi: 10.1007/BF02783432. [17] 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. [18] 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. [19] D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices, Math. Res. Lett., 9 (2002), 363-378. [20] 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. [21] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergod. Th. Dynam. Sys., 17 (1997), 147-167. doi: 10.1017/S0143385797060987. [22] M. Kesseböhmer, Large deviation for weak Gibbs measures and multifractal spectra, Nonlinearity, 14 (2001), 395-409. doi: 10.1088/0951-7715/14/2/312. [23] M. Kesseböhmer and B. Stratmann, A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups, Ergod. Th. Dynam. Sys., 24 (2004), 141-170. [24] N. G. Makarov, Fine structure of harmonic measure, St. Peterburg Math. J., 10 (1999), 217-268. [25] A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Contin. Dyn. Syst., 16 (2006), 435-454. doi: 10.3934/dcds.2006.16.435. [26] E. Olivier, Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for $g$-measures, Nonlinearity, 12 (1999), 1571-1585. doi: 10.1088/0951-7715/12/6/309. [27] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649. doi: 10.1016/j.matpur.2003.09.007. [28] Y. Peres, M. Rams, K. Simon and B. Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc., 129 (2001), 2689-2699. doi: 10.1090/S0002-9939-01-05969-X. [29] Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. [30] 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. [31] 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. [32] D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergod. Th. Dynam. Sys., 9 (1989), 527-541. [33] 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.

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##### References:
 [1] J. Barral and D. J. Feng, Weighted thermodynamic formalism and applications, arXiv:0909.4247v1. [2] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. Dynam. Sys., 16 (1996), 871-927. doi: 10.1017/S0143385700010117. [3] L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Contin. Dyn. Syst., 16 (2006), 279-305. doi: 10.3934/dcds.2006.16.279. [4] L. Barreira and P. Doutor, Almost additive multifractal analysis, J. Math. Pures Appl., 92 (2009), 1-17. doi: 10.1016/j.matpur.2009.04.006. [5] L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214 (2000), 339-371. doi: 10.1007/s002200000268. [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] R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. [9] R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J., 7 (1940), 312-340. doi: 10.1215/S0012-7094-40-00718-9. [10] G. Brown, G. Michon and J. Peyrière, On the multifractal analysis of measures, J. Stat. Phys., 66 (1992), 775-790. doi: 10.1007/BF01055700. [11] Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657. [12] P. Collet, J. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Statist. Phys., 47 (1987), 609-644. doi: 10.1007/BF01206149. [13] 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. [14] 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. [15] A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244. doi: 10.1017/S0024610701002137. [16] D.-J. Feng, Lyapounov exponents for products of matrices and multifractal analysis. Part I: Positive matrices, Israel J. Math., 138 (2003), 353-376. doi: 10.1007/BF02783432. [17] 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. [18] 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. [19] D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices, Math. Res. Lett., 9 (2002), 363-378. [20] 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. [21] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergod. Th. Dynam. Sys., 17 (1997), 147-167. doi: 10.1017/S0143385797060987. [22] M. Kesseböhmer, Large deviation for weak Gibbs measures and multifractal spectra, Nonlinearity, 14 (2001), 395-409. doi: 10.1088/0951-7715/14/2/312. [23] M. Kesseböhmer and B. Stratmann, A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups, Ergod. Th. Dynam. Sys., 24 (2004), 141-170. [24] N. G. Makarov, Fine structure of harmonic measure, St. Peterburg Math. J., 10 (1999), 217-268. [25] A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Contin. Dyn. Syst., 16 (2006), 435-454. doi: 10.3934/dcds.2006.16.435. [26] E. Olivier, Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for $g$-measures, Nonlinearity, 12 (1999), 1571-1585. doi: 10.1088/0951-7715/12/6/309. [27] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649. doi: 10.1016/j.matpur.2003.09.007. [28] Y. Peres, M. Rams, K. Simon and B. Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc., 129 (2001), 2689-2699. doi: 10.1090/S0002-9939-01-05969-X. [29] Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. [30] 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. [31] 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. [32] D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergod. Th. Dynam. Sys., 9 (1989), 527-541. [33] 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.
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