no. 2
Transience and multifractal analysis
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 407-421.

We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.

DOI : 10.1016/j.anihpc.2015.12.007
Mots clés : Multifractal analysis, Ergodic theory, Lyapunov exponents 
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Iommi, Godofredo; Jordan, Thomas; Todd, Mike. Transience and multifractal analysis. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 407-421. doi : 10.1016/j.anihpc.2015.12.007. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.12.007/

[1] Avila, A.; Lyubich, M. Hausdorff dimension and conformal measures of Feigenbaum Julia sets, J. Am. Math. Soc., Volume 21 (2008) no. 2, pp. 305–363 | DOI | MR | Zbl

[2] Barreira, L. Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, vol. 272, Birkhauser Verlag, Basel, 2008 | MR | Zbl

[3] Barreira, L.; Saussol, B. Variational principles and mixed multifractal spectra, Trans. Am. Math. Soc., Volume 353 (2001) no. 10, pp. 3919–3944 | DOI | MR | Zbl

[4] Barreira, L.; Saussol, B.; Schmeling, J. Higher-dimensional multifractal analysis, J. Math. Pures Appl. (9), Volume 81 (2002), pp. 67–91 | DOI | MR | Zbl

[5] Barreira, L.; Schmeling, J. Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Isr. J. Math., Volume 116 (2000), pp. 29–70 | DOI | MR | Zbl

[6] Bruin, H.; Todd, M. Transience and thermodynamic formalism for infinitely branched interval maps, J. Lond. Math. Soc., Volume 86 (2012), pp. 171–194 | DOI | MR | Zbl

[7] Bruin, H.; Todd, M. Wild attractors and thermodynamic formalism, Monatshefte Math., Volume 178 (2015), pp. 39–83 | DOI | MR | Zbl

[8] Climenhaga, V. The thermodynamic approach to multifractal analysis, Ergod. Theory Dyn. Syst., Volume 34 (2014) no. 5, pp. 1409–1450 | DOI | MR | Zbl

[9] Cyr, V. Countable Markov shifts with transient potentials, Proc. Lond. Math. Soc., Volume 103 (2011), pp. 923–949 | MR | Zbl

[10] Cyr, V.; Sarig, O. Spectral gap and transience for Ruelle operators on countable Markov shifts, Commun. Math. Phys., Volume 292 (2009), pp. 637–666 | MR | Zbl

[11] Falconer, K. Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Inc., Hoboken, NJ, 2003 | MR

[12] Falk, K.; Stratmann, B. Remarks on Hausdorff dimensions for transient limit sets of Kleinian groups, Tohoku Math. J. (2), Volume 56 (2004) no. 4, pp. 571–582 | DOI | MR | Zbl

[13] Fan, A.; Feng, D.; Wu, J. Recurrence, dimension and entropy, J. Lond. Math. Soc. (2), Volume 64 (2001) no. 1, pp. 229–244 | MR | Zbl

[14] Fan, A.; Liao, L.; Peyriére, J. Generic points in systems of specification and Banach valued Birkhoff ergodic average, Discrete Contin. Dyn. Syst., Volume 21 (2008) no. 4, pp. 1103–1128 | MR | Zbl

[15] Fan, A.; Liao, L.; Wang, B.; Wu, J. On Khintchine exponents and Lyapunov exponents of continued fractions, Ergod. Theory Dyn. Syst., Volume 29 (2009) no. 1, pp. 73–109 | MR | Zbl

[16] Feng, D.; Lau, K.-S.; Wu, J. Ergodic limits on the conformal repellers, Adv. Math., Volume 169 (2002) no. 1, pp. 58–91 | DOI | MR | Zbl

[17] Gelfert, K.; Rams, M. The Lyapunov spectrum of some parabolic systems, Ergod. Theory Dyn. Syst., Volume 29 (2009) no. 3, pp. 919–940 | DOI | MR | Zbl

[18] Gurevič, B.M. Topological entropy for denumerable Markov chains, Dokl. Akad. Nauk SSSR, Volume 10 (1969), pp. 911–915 | Zbl

[19] Hanus, P.; Mauldin, R.D.; Urbanski, M. Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems, Acta Math. Hung., Volume 96 (2002) no. 1–2, pp. 27–98 | MR | Zbl

[20] Hofbauer, F. Multifractal spectra of Birkhoff averages for a piecewise monotone interval map, Fundam. Math., Volume 208 (2010) no. 2, pp. 95–121 | DOI | MR | Zbl

[21] Hofbauer, F.; Raith, P. The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval, Can. Math. Bull., Volume 35 (1992) no. 1, pp. 84–98 | DOI | MR | Zbl

[22] Iommi, G. Multifractal analysis for countable Markov shifts, Ergod. Theory Dyn. Syst., Volume 25 (2005), pp. 1881–1907 | DOI | MR | Zbl

[23] Iommi, G.; Jordan, T. Multifractal analysis of Birkhoff averages for countable Markov maps, Ergod. Theory Dyn. Syst., Volume 35 (2015) no. 8, pp. 2559–2586 | DOI | MR | Zbl

[24] Iommi, G.; Jordan, T. Multifractal analysis of quotients of Birkhoff sums for countable Markov maps, Int. Math. Res. Not., Volume 2 (2015), pp. 460–498 | MR | Zbl

[25] Iommi, G.; Jordan, T.; Todd, M. Recurrence and transience for suspension flows, Isr. J. Math., Volume 209 (2015) no. 2, pp. 547–592 | DOI | MR | Zbl

[26] Iommi, G.; Todd, M. Transience in dynamical systems, Ergod. Theory Dyn. Syst., Volume 33 (2013) no. 5, pp. 1450–1476 | DOI | MR | Zbl

[27] Johansson, A.; Jordan, T.; Oberg, A.; Pollicott, M. Multifractal analysis of non-uniformly hyperbolic systems, Isr. J. Math., Volume 177 (2010), pp. 125–144 | DOI | MR | Zbl

[28] Kesseböhmer, M.; Munday, S.; Stratmann, B. Strong renewal theorems and Lyapunov spectra for a -Farey and a -Lüroth systems, Ergod. Theory Dyn. Syst., Volume 32 (2012) no. 3, pp. 989–1017 | DOI | MR | Zbl

[29] Kesseböhmer, M.; Stratmann, B. A multifractal analysis for Stern–Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math., Volume 605 (2007), pp. 133–163 | MR | Zbl

[30] Kesseböhmer, M.; Urbański, M. Higher-dimensional multifractal value sets for conformal infinite graph directed Markov systems, Nonlinearity, Volume 20 (2007) no. 8, pp. 1969–1985 | DOI | MR | Zbl

[31] Makarov, N.; Smirnov, S. On “thermodynamics” of rational maps. I. Negative spectrum, Commun. Math. Phys., Volume 211 (2000) no. 3, pp. 705–743 | DOI | MR | Zbl

[32] Manning, A. A relation between Lyapunov exponents, Hausdorff dimension and entropy, Ergod. Theory Dyn. Syst., Volume 1 (1981) no. 4, pp. 451–459 | DOI | MR | Zbl

[33] Mauldin, R.; Urbański, M. Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003 | MR | Zbl

[34] Nakaishi, K. Multifractal formalism for some parabolic maps, Ergod. Theory Dyn. Syst., Volume 20 (2000) no. 3, pp. 843–857 | DOI | MR | Zbl

[35] Olivier, E. Structure multifractale d'une dynamique non-expansive définie sur un ensemble de Cantor, C. R. Acad. Sci. Paris Sér. I Math., Volume 331 (2000) no. 8, pp. 605–610 | DOI | MR | Zbl

[36] Olsen, L. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. (9), Volume 82 (2003) no. 12, pp. 1591–1649 | DOI | MR | Zbl

[37] Patterson, S.J. Further remarks on the exponent of convergence of Poincaré series, Tohoku Math. J. (2), Volume 35 (1983) no. 3, pp. 357–373 | DOI | MR | Zbl

[38] Pesin, Y.; Weiss, H. The multifractal analysis of Birkhoff averages and large deviations, Global Analysis of Dynamical Systems, Inst. Phys., Bristol, 2001, pp. 419–431 | MR | Zbl

[39] Pollicott, M.; Weiss, H. Multifractal analysis of Lyapunov exponent for continued fraction and Manneville–Pomeau transformations and applications to Diophantine approximation, Commun. Math. Phys., Volume 207 (1999) no. 1, pp. 145–171 | DOI | MR | Zbl

[40] Przytycki, F.; Urbański, M. Conformal Fractals: Ergodic Theory Methods, Cambridge University Press, 2010 | DOI

[41] Sarig, O. Thermodynamic formalism for countable Markov shifts, Ergod. Theory Dyn. Syst., Volume 19 (1999), pp. 1565–1593 | DOI | MR | Zbl

[42] Sarig, O. Phase transitions for countable Markov shifts, Commun. Math. Phys., Volume 217 (2001) no. 3, pp. 555–577 | DOI | MR | Zbl

[43] Sarig, O. Existence of Gibbs measures for countable Markov shifts, Proc. Am. Math. Soc., Volume 131 (2003), pp. 1751–1758 | DOI | MR | Zbl

[44] Stratmann, B.; Vogt, R. Fractal dimension of dissipative sets, Nonlinearity, Volume 10 (1997), pp. 565–577 | DOI | MR | Zbl

[45] Stratmann, B.; Urbański, M. Pseudo-Markov systems and infinitely generated Schottky groups, Am. J. Math., Volume 129 (2007) no. 4, pp. 1019–1062 | DOI | MR | Zbl

[46] Takens, F.; Verbitskiy, E. On the variational principle for the topological entropy of certain non-compact sets, Ergod. Theory Dyn. Syst., Volume 23 (2003) no. 1, pp. 317–348 | DOI | MR | Zbl

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