# 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 & Continuous Dynamical Systems - A, 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}., (). Google Scholar [2] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. Dynam. Sys., 16 (1996), 871. doi: 10.1017/S0143385700010117. Google Scholar [3] 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 [4] L. Barreira and P. Doutor, Almost additive multifractal analysis,, J. Math. Pures Appl., 92 (2009), 1. doi: 10.1016/j.matpur.2009.04.006. Google Scholar [5] L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows,, Comm. Math. Phys., 214 (2000), 339. doi: 10.1007/s002200000268. 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] R. Bowen, Hausdorff dimension of quasicircles,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11. Google Scholar [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. doi: 10.1215/S0012-7094-40-00718-9. Google Scholar [10] G. Brown, G. Michon and J. Peyrière, On the multifractal analysis of measures,, J. Stat. Phys., 66 (1992), 775. doi: 10.1007/BF01055700. Google Scholar [11] 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 [12] P. Collet, J. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems,, J. Statist. Phys., 47 (1987), 609. doi: 10.1007/BF01206149. Google Scholar [13] 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 [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. doi: 10.1023/A:1018643512559. Google Scholar [15] A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy,, J. London Math. Soc., 64 (2001), 229. doi: 10.1017/S0024610701002137. Google Scholar [16] D.-J. Feng, Lyapounov exponents for products of matrices and multifractal analysis. Part I: Positive matrices,, Israel J. Math., 138 (2003), 353. doi: 10.1007/BF02783432. Google Scholar [17] 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 [18] 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 [19] D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices,, Math. Res. Lett., 9 (2002), 363. Google Scholar [20] 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 [21] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps,, Ergod. Th. Dynam. Sys., 17 (1997), 147. doi: 10.1017/S0143385797060987. Google Scholar [22] M. Kesseböhmer, Large deviation for weak Gibbs measures and multifractal spectra,, Nonlinearity, 14 (2001), 395. doi: 10.1088/0951-7715/14/2/312. Google Scholar [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. Google Scholar [24] N. G. Makarov, Fine structure of harmonic measure,, St. Peterburg Math. J., 10 (1999), 217. Google Scholar [25] 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 [26] E. Olivier, Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for $g$-measures,, Nonlinearity, 12 (1999), 1571. doi: 10.1088/0951-7715/12/6/309. Google Scholar [27] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages,, J. Math. Pures Appl., 82 (2003), 1591. doi: 10.1016/j.matpur.2003.09.007. Google Scholar [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. doi: 10.1090/S0002-9939-01-05969-X. Google Scholar [29] Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications,", Chicago Lectures in Mathematics, (1997). Google Scholar [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. doi: 10.1007/BF02180206. Google Scholar [31] 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 [32] D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters,, Ergod. Th. Dynam. Sys., 9 (1989), 527. Google Scholar [33] D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Encyclopedia of Mathematics and its Applications, 5 (1978). Google Scholar

show all references

##### References:
 [1] J. Barral and D. J. Feng, Weighted thermodynamic formalism and applications,, \arXiv{0909.4247v1}., (). Google Scholar [2] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. Dynam. Sys., 16 (1996), 871. doi: 10.1017/S0143385700010117. Google Scholar [3] 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 [4] L. Barreira and P. Doutor, Almost additive multifractal analysis,, J. Math. Pures Appl., 92 (2009), 1. doi: 10.1016/j.matpur.2009.04.006. Google Scholar [5] L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows,, Comm. Math. Phys., 214 (2000), 339. doi: 10.1007/s002200000268. 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] R. Bowen, Hausdorff dimension of quasicircles,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11. Google Scholar [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. doi: 10.1215/S0012-7094-40-00718-9. Google Scholar [10] G. Brown, G. Michon and J. Peyrière, On the multifractal analysis of measures,, J. Stat. Phys., 66 (1992), 775. doi: 10.1007/BF01055700. Google Scholar [11] 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 [12] P. Collet, J. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems,, J. Statist. Phys., 47 (1987), 609. doi: 10.1007/BF01206149. Google Scholar [13] 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 [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. doi: 10.1023/A:1018643512559. Google Scholar [15] A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy,, J. London Math. Soc., 64 (2001), 229. doi: 10.1017/S0024610701002137. Google Scholar [16] D.-J. Feng, Lyapounov exponents for products of matrices and multifractal analysis. Part I: Positive matrices,, Israel J. Math., 138 (2003), 353. doi: 10.1007/BF02783432. Google Scholar [17] 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 [18] 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 [19] D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices,, Math. Res. Lett., 9 (2002), 363. Google Scholar [20] 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 [21] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps,, Ergod. Th. Dynam. Sys., 17 (1997), 147. doi: 10.1017/S0143385797060987. Google Scholar [22] M. Kesseböhmer, Large deviation for weak Gibbs measures and multifractal spectra,, Nonlinearity, 14 (2001), 395. doi: 10.1088/0951-7715/14/2/312. Google Scholar [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. Google Scholar [24] N. G. Makarov, Fine structure of harmonic measure,, St. Peterburg Math. J., 10 (1999), 217. Google Scholar [25] 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 [26] E. Olivier, Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for $g$-measures,, Nonlinearity, 12 (1999), 1571. doi: 10.1088/0951-7715/12/6/309. Google Scholar [27] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages,, J. Math. Pures Appl., 82 (2003), 1591. doi: 10.1016/j.matpur.2003.09.007. Google Scholar [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. doi: 10.1090/S0002-9939-01-05969-X. Google Scholar [29] Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications,", Chicago Lectures in Mathematics, (1997). Google Scholar [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. doi: 10.1007/BF02180206. Google Scholar [31] 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 [32] D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters,, Ergod. Th. Dynam. Sys., 9 (1989), 527. Google Scholar [33] D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Encyclopedia of Mathematics and its Applications, 5 (1978). Google Scholar
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