Localized asymptotic  behavior for almost additive potentials

Julien Barral; Yan-Hui Qu

doi:10.3934/dcds.2012.32.717

Article Contents

2012, Volume 32, Issue 3: 717-751. Doi: 10.3934/dcds.2012.32.717

This issue Previous Article On the Cauchy problem for nonlinear Schrödinger equations with rotation Next Article Instability for a priori unstable Hamiltonian systems: A dynamical approach

Localized asymptotic behavior for almost additive potentials

Julien Barral^1, and
Yan-Hui Qu^2,

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

Received: September 2010

Revised: December 2010

Published online: October 2011

Abstract / Introduction Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: Primary: 37B40; Secondary: 28A80.

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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.