Inventiones mathematicae

, Volume 218, Issue 3, pp 853–888 | Cite as

Large deviation principle in one-dimensional dynamics

  • Yong Moo ChungEmail author
  • Juan Rivera-Letelier
  • Hiroki Takahasi


We study the dynamics of smooth interval maps with non-flat critical points. For every such a map that is topologically exact, we establish the full (level-2) Large Deviation Principle for empirical means. In particular, the Large Deviation Principle holds for every non-renormalizable quadratic map. This includes the maps without physical measure found by Hofbauer and Keller, and challenges the widely-shared view of the Large Deviation Principle as a refinement of laws of large numbers.

Mathematics Subject Classification

37A50 37C40 37D25 37D45 37E05 



We would like to thank Michał Misiurewicz for his help with references, Bing Gao, Gerhard Keller and Masato Tsujii for fruitful discussions, and the anonymous referees for their healthy criticism that helped us improve the exposition in the introduction. The first-named author is partially supported by the Grant-in-Aid for Scientific Research (C) of the JSPS 16K05179. The second-named author is partially supported by FONDECYT Grant 1141091 and NSF Grant DMS-1700291. The last-named author is partially supported by the Grant-in-Aid for Young Scientists (A) of the JSPS 15H05435 and the Grant-in-Aid for Scientific Research (B) of the JSPS 16KT0021.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Yong Moo Chung
    • 1
    Email author
  • Juan Rivera-Letelier
    • 2
  • Hiroki Takahasi
    • 3
  1. 1.Department of Applied MathematicsHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Department of MathematicsUniversity of RochesterRochesterUSA
  3. 3.Department of Mathematics, Keio Institute of Pure and Applied Sciences (KiPAS)Keio UniversityYokohamaJapan

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