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Multifractal analysis of weak Gibbs measures for non-uniformly expanding C1 maps

Published online by Cambridge University Press:  18 January 2010

THOMAS JORDAN  and
MICHAŁ RAMS
THOMAS JORDAN
Affiliation:
Department of Mathematics, The University of Bristol, University Walk, Clifton, Bristol BS8 1TW, UK (email: thomas.jordan@bristol.ac.uk)
MICHAŁ RAMS
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland (email: M.Rams@impan.gov.pl)
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Abstract

We will consider the local dimension spectrum of a weak Gibbs measure on a C1 non-uniformly hyperbolic system of Manneville–Pomeau type. We will present the spectrum in three ways: using invariant measures, ergodic invariant measures supported on hyperbolic sets and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 1 , February 2011 , pp. 143 - 164
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Barreira, L. and Saussol, B.. Variational principles and mixed multifractal spectra. Trans. Amer. Math. Soc. 353 (2001), 3919–3944.CrossRefGoogle Scholar
[2]Byrne, W.. Multifractal analysis of parabolic rational maps. PhD Thesis, The University of North Texas.Google Scholar
[3]Cawley, R. and Mauldin, D.. Multifractal decompositions of Moran fractals. Adv. Math. 92(2) (1992), 196–236.CrossRefGoogle Scholar
[4]Contreras, G., Lopes, A. O. and Thieullen, Ph.. Lyapunov minimizing measures for expanding maps of the circle. Ergod. Th. & Dynam. Sys. 21 (2001), 1379–1409.CrossRefGoogle Scholar
[5]Gelfert, K. and Rams, M.. Geometry of limit set for expansive Markov systems. Trans. Amer. Math. Soc. 361 (2009), 2001–2020.CrossRefGoogle Scholar
[6]Gelfert, K. and Rams, M.. Multifractal analysis of Lyapunov exponents of parabolic iterated function systems. Ergod. Th. & Dynam. Sys. 29 (2009), 919–940.CrossRefGoogle Scholar
[7]Hanus, P., Mauldin, R. and Urbański, M.. Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems. Acta Math. Hungar. 96 (2002), 27–98.CrossRefGoogle Scholar
[8]Hofbauer, F. and Raith, P.. The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval. Canad. Math. Bull. 35(1) (1992), 84–98.CrossRefGoogle Scholar
[9]Johansson, A., Jordan, T., Öberg, A. and Pollicott, M.. Multifractal analysis of non-uniformly hyperbolic systems. Israel J. Math. to appear, 2008. Preprint available atwww.maths.bris.ac.uk/∼matmj/atam103.ps.Google Scholar
[10]Kesseböhmer, M.. Large deviation for weak Gibbs measures and multifractal spectra. Nonlinearity 14 (2001), 395–409.CrossRefGoogle Scholar
[11]Kesseböhmer, M. and Stratmann, B.. A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups. Ergod. Th. & Dynam. Sys. 24 (2004), 141–170.CrossRefGoogle Scholar
[12]Kesseböhmer, M. and Stratmann, B.. A multifractal analysis for Stern–Brocot intervals, continued fractions and Diophantine growth rates. J. Reine Angew. Math. 605 (2007), 133–163.Google Scholar
[13]Ledrappier, F.. Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. & Dynam. Sys. 1 (1981), 77–93.CrossRefGoogle Scholar
[14]Nakaishi, K.. Multifractal formalism for some parabolic maps. Ergod. Th. & Dynam. Sys. 20 (2000), 843–857.CrossRefGoogle Scholar
[15]Olsen, L.. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl. (9) 82 (2003), 1591–1649.CrossRefGoogle Scholar
[16]Pesin, Y.. Dimension theory in dynamical systems. Contemporary Views and Applications (Chicago Lectures in Mathematics). University of Chicago Press, Chicago, 1997.Google Scholar
[17]Pesin, Y. and Weiss, H.. The multifractal analysis of Birkhoff averages and large deviations. Global Analysis of Dynamical Systems. Institute of Physics, Bristol, 2001, pp. 419–431.Google Scholar
[18]Rand, D.. The singularity spectrum f(α) for cookie-cutters. Ergod. Th. & Dynam. Sys. 9(3) (1989), 527–541.CrossRefGoogle Scholar
[19]Stratmann, B. and Urbański, M.. Real analyticity of topological pressure for parabolically semihyperbolic generalized polynomial-like maps. Indag. Math. 14 (2003), 119–134.CrossRefGoogle Scholar
[20]Stratmann, B. and Urbański, M.. Multifractal analysis for parabolically semihyperbolic generalized polynomial-like maps. Complex Dynamics and Related Topics (New Studies in Advanced Mathematics). Eds. Jiang, Y. and Wang, Y.. International Press of Boston, Boston, 2003, pp. 393–447.Google Scholar
[21]Urbański, M.. Parabolic Cantor sets. Fund. Math. 151 (1996), 241–277.Google Scholar
[22]Walters, P.. An Introduction to Ergodic Theory. Springer, Berlin, 1982.CrossRefGoogle Scholar
[23]Yuri, M.. Weak Gibbs measures for certain non-hyperbolic systems. Ergod. Th. & Dynam. Sys. 20 (2000), 1495–1518.CrossRefGoogle Scholar
[24]Yuri, M.. Multifractal analysis of weak Gibbs measures for intermittent systems. Comm. Math. Phys. 230(2) (2002), 365–388.CrossRefGoogle Scholar