Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-28T17:23:09.627Z Has data issue: false hasContentIssue false

Multifractal formalism for Benedicks–Carleson quadratic maps

Published online by Cambridge University Press:  11 March 2013

YONG MOO CHUNG
Affiliation:
Department of Applied Mathematics, Hiroshima University, Higashi-Hiroshima 739-8527, Japan email chung@amath.hiroshima-u.ac.jp
HIROKI TAKAHASI
Affiliation:
Department of Electronic Science and Engineering, Kyoto University, Kyoto 606-8501, Japan email takahasi.hiroki.7r@kyoto-u.ac.jp

Abstract

For a positive measure set of non-uniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given continuous function and consider the associated Birkhoff spectrum which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to ‘see’ sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle of empirical distributions, with Lebesgue as a reference measure.

Type
Research Article
Copyright
Copyright ©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barreira, L. and Schmeling, J.. Sets of ‘non-typical’ points have full Hausdorff dimension and full topological entropy. Israel J. Math. 116 (2000), 2970.Google Scholar
Benedicks, M. and Carleson, L.. On iterations of $1- a{x}^{2} $ on $(- 1, 1)$. Ann. Math. 122 (1985), 125.Google Scholar
Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. Math. 133 (1991), 73169.Google Scholar
Benedicks, M. and Young, L.-S.. Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergod. Th. & Dynam. Sys. 12 (1992), 1337.Google Scholar
Bruin, H. and Keller, G.. Equilibrium states for S-unimodal maps. Ergod. Th. & Dynam. Sys. 18 (1998), 765789.Google Scholar
Chung, Y. M.. Birkhoff spectra for one-dimensional maps with some hyperbolicity. Stoch. Dyn. 10 (2010), 5375.Google Scholar
Chung, Y. M.. Large deviations on Markov towers. Nonlinearity 24 (2011), 12291252.Google Scholar
Chung, Y. M. and Takahasi, H.. Large deviation principle for Benedicks–Carleson quadratic maps. Comm. Math. Phys. 315 (2012), 803826.Google Scholar
Climenhaga, V.. Bowen’s equation in the non-uniform setting. Ergod. Th. & Dynam. Sys. 31 (2011), 11631182.Google Scholar
Comman, H. and Rivera-Letelier, J.. Large deviation principles for non-uniformly hyperbolic rational maps. Ergod. Th. & Dynam. Sys. 31 (2011), 321349.Google Scholar
de Melo, W. and van Strien, S.. One-dimensional Dynamics. Springer, Berlin, 1993.Google Scholar
Dembo, A. and Zeitouni, O.. Large Deviations Techniques and Applications (Applications of Mathematics, 38), 2nd edn. Springer, New York, 1998.Google Scholar
Gelfert, K., Przytycki, F. and Rams, M.. On the Lyapunov spectrum for rational maps. Math. Ann. 348 (2010), 9651004.Google Scholar
Grassberger, P., Badii, R. and Politi, A.. Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51 (1988), 135178.Google Scholar
Hasley, T. C., Jensen, M., Kadanoff, L., Procaccia, I. and Shraiman, B.. Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33 (2) (1986), 11411151.Google Scholar
Hofbauer, F.. Local dimension for piecewise monotone maps on the interval. Ergod. Th. & Dynam. Sys. 15 (1995), 11191142.Google Scholar
Iommi, G. and Todd, M.. Dimension theory for multimodal maps. Ann. Henri Poincaré 12 (2011), 591620.Google Scholar
Jakobson, M.. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 81 (1981), 3988.CrossRefGoogle Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Étud. Sci. 51 (1980), 137173.Google Scholar
Keller, G. and Nowicki, T.. Spectral theory, zeta functions and the distribution of periodic points for Collet–Eckmann maps. Comm. Math. Phys. 149 (1992), 3169.Google Scholar
Kifer, Y.. Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc. 321 (1990), 505524.Google Scholar
Ledrappier, F.. Some properties of absolutely continuous invariant measures of an interval. Ergod. Th. & Dynam. Sys. 1 (1981), 7793.Google Scholar
Ledrappier, F.. Some relations between dimension and Lyapunov exponents. Comm. Math. Phys. 81 (1981), 229238.Google Scholar
Nakaishi, K.. Multifractal formalism for some parabolic maps. Ergod. Th. & Dynam. Sys. 20 (2000), 843857.CrossRefGoogle Scholar
Nowicki, T. and Sands, D.. Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132 (1998), 633680.Google Scholar
Olsen, L.. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl. 82 (2003), 15911649.Google Scholar
Orey, S. and Pelikan, S.. Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms. Trans. Amer. Math. Soc. 315 (1989), 741753.Google Scholar
Pesin, Y.. Dimension Theory in Dynamical Systems. University of Chicago Press, Chicago, IL, 1997.Google Scholar
Pesin, Y. and Weiss, H.. A multifracatal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J. Stat. Phys. 86 (1997), 233275.CrossRefGoogle Scholar
Pesin, Y. and Weiss, H.. The multifractal analysis of Birkhoff averages and large deviations. Global Analysis of Dynamical Systems. Eds. Broer, H. W., Krauskopf, B. and Vegter, G.. Institute of Physics, Bristol, UK, 2001, pp. 419431.Google Scholar
Przytycki, F. and Rivera-Letelier, J.. Nice inducing schemes and the thermodynamics of rational maps. Comm. Math. Phys. 70 (2011), 661707.Google Scholar
Rey-Bellet, L. and Young, L.-S.. Large deviations in non-uniformly hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 28 (2008), 587612.Google Scholar
Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Mat. Bras. 9 (1978), 8387.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar
Weiss, H.. The Lyapunov spectrum for conformal expanding maps and Axiom A surface diffeomorphisms. J. Stat. Phys. 95 (1999), 615632.Google Scholar
Young, L.-S.. Decay of correlations of certain quadratic maps. Comm. Math. Phys. 146 (1992), 123138.Google Scholar
Young, L.-S.. Dimension, entropy and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 2 (1982), 109124.Google Scholar