Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-14T03:58:37.598Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviation principles of one-dimensional maps for Hölder continuous potentials

Published online by Cambridge University Press:  04 August 2014

HUAIBIN LI
HUAIBIN LI*
Affiliation:
School of Mathematics and Information Science, Henan University, Kaifeng 475004, PR China email matlihb@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We show some level-2 large deviation principles for real and complex one-dimensional maps satisfying a weak form of hyperbolicity. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 1 , February 2016, pp. 127 - 141
Copyright
© Cambridge University Press, 2014 

References

Araújo, V. and Pacifico, M. J.. Large deviations for non-uniformly expanding maps. J. Stat. Phys. 125(2) (2006), 415–457.CrossRefGoogle Scholar
Bruin, H. and Keller, G.. Equilibrium states for S-unimodal maps. Ergod. Th. & Dynam. Sys. 18(4) (1998), 765–789.CrossRefGoogle Scholar
Bruin, H., Rivera-Letelier, J., Shen, W. and van Strien, S.. Large derivatives, backward contraction and invariant densities for interval maps. Invent. Math. 172(3) (2008), 509–533.CrossRefGoogle Scholar
Carleson, L. and Gamelin, T. W.. Complex Dynamics (Universitext: Tracts in Mathematics). Springer, New York, 1993.CrossRefGoogle Scholar
Comman, H. and Rivera-Letelier, J.. Large deviation principles for non-uniformly hyperbolic rational maps. Ergod. Th. & Dynam. Sys. 31(2) (2011), 321–349.CrossRefGoogle Scholar
Chung, Y. M. and Takahasi, H.. Large deviation principle for Benedicks–Carleson quadratic maps. Comm. Math. Phys. 315(3) (2012), 803–826.CrossRefGoogle Scholar
Climenhaga, V., Thompson, D. J. and Yamamoto, K.. Large deviations for systems with non-uniform structure. Preprint, 2013, arXiv:1304.5497v1.Google Scholar
Denker, M.. Probability theory for rational maps. Probability Theory and Mathematical Statistics (St. Petersburg, 1993). Gordon and Breach, Amsterdam, 1996, pp. 29–40.Google Scholar
de Melo, W. and van Strien, S.. One-dimensional Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25). Springer, Berlin, 1993.Google Scholar
Dobbs, N.. On cusps and flat tops. Preprint, 2008, arXiv:0801.3815v2.Google Scholar
Dobbs, N.. Pesin theory and equilibrium measures on the interval. Preprint, 2013,arXiv:1304.3305v1.Google Scholar
Denker, M. and Urbański, M.. Ergodic theory of equilibrium states for rational maps. Nonlinearity 4(1) (1991), 103–134.CrossRefGoogle Scholar
Inoquio-Renteria, I. and Rivera-Letelier, J.. A characterization of hyperbolic potentials of rational maps. Bull. Braz. Math. Soc. (N.S.) 43(1) (2012), 99–127.CrossRefGoogle Scholar
Keller, G.. Equilibrium States in Ergodic Theory (London Mathematical Society Student Texts, 42). Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
Kifer, Y.. Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc. 321(2) (1990), 505–524.CrossRefGoogle Scholar
Keller, G. and Nowicki, T.. Spectral theory, zeta functions and the distribution of periodic points for Collet–Eckmann maps. Comm. Math. Phys. 149(1) (1992), 31–69.CrossRefGoogle Scholar
Kozlovski, O. and van Strien, S.. Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials. Proc. Lond. Math. Soc. (3) 99(2) (2009), 275–296.CrossRefGoogle Scholar
Li, H. and Rivera-Letelier, J.. Equilibrium states of interval maps for hyperbolic potentials. Nonlinearity to appear. Preprint, 2014, arXiv:1210.6952v3 [math.DS].CrossRefGoogle Scholar
Li, H. and Rivera-Letelier, J.. Equilibrium states of weakly hyperbolic one-dimensional maps for Hölder potentials. Comm. Math. Phys. 328(1) (2014), 397–419.CrossRefGoogle Scholar
Li, H. and Shen, W.. On non-uniform hyperbolicity assumptions in one-dimensional dynamics. Sci. China Math. 53(7) (2010), 1663–1677.CrossRefGoogle Scholar
Melbourne, I.. Large and moderate deviations for slowly mixing dynamical systems. Proc. Amer. Math. Soc. 137(5) (2009), 1735–1741.CrossRefGoogle Scholar
Milnor, J.. Dynamics in One Complex Variable (Annals of Mathematics Studies, 160). 3rd edn. Princeton University Press, Princeton, NJ, 2006.Google Scholar
Melbourne, I. and Nicol, M.. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360(12) (2008), 6661–6676.CrossRefGoogle Scholar
Przytycki, F.. On the Perron–Frobenius–Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions. Bol. Soc. Brasil. Mat. (N.S.) 20(2) (1990), 95–125.CrossRefGoogle Scholar
Pollicott, M. and Sharp, R.. Large deviations and the distribution of pre-images of rational maps. Comm. Math. Phys. 181(3) (1996), 733–739.CrossRefGoogle Scholar
Pollicott, M. and Sharp, R.. Large deviations, fluctuations and shrinking intervals. Comm. Math. Phys. 290(1) (2009), 321–334.CrossRefGoogle Scholar
Pollicott, M. and Sharp, R.. Large deviations for intermittent maps. Nonlinearity 22(9) (2009), 2079–2092.CrossRefGoogle Scholar
Pollicott, M., Sharp, R. and Yuri, M.. Large deviations for maps with indifferent fixed points. Nonlinearity 11(4) (1998), 1173–1184.CrossRefGoogle Scholar
Przytycki, F. and Urbański, M.. Conformal Fractals: Ergodic Theory Methods (London Mathematical Society Lecture Note Series, 371). Cambridge University Press, Cambridge, 2010.CrossRefGoogle Scholar
Rey-Bellet, L. and Young, L.-S.. Large deviations in non-uniformly hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 28(2) (2008), 587–612.CrossRefGoogle Scholar
Rivera-Letelier, J.. A connecting lemma for rational maps satisfying a no-growth condition. Ergod. Th. & Dynam. Sys. 27(2) (2007), 595–636.CrossRefGoogle Scholar
Rivera-Letelier, J.. On the asymptotic expansion of maps with disconnected Julia set. Preprint, 2012, arXiv:1206.2376v1.Google Scholar
Rivera-Letelier, J. and Shen, W.. Statistical properties of one-dimensional maps under weak hyperbolicity assumptions. Ann. Sci. École Norm. Sup. to appear. Preprint, 2010,arXiv:1004.0230v1.Google Scholar
Xia, H. and Fu, X.. Remarks on large deviation for rational maps on the Riemann sphere. Stoch. Dyn. 7(3) (2007), 357–363.CrossRefGoogle Scholar
Young, L.-S.. Large deviations in dynamical systems. Trans. Amer. Math. Soc. 318(2) (1990), 525–543.Google Scholar