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Science China Mathematics

, Volume 61, Issue 12, pp 2299–2310 | Cite as

The local Hölder exponent for the entropy of real unimodal maps

  • Giulio TiozzoEmail author
Articles
  • 19 Downloads

Abstract

We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ (equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is locally Hölder continuous where h(θ) > 0, and more precisely for any θ which does not lie in a plateau the local Hölder exponent equals exactly, up to a factor log 2, the value of the function at that point. This confirms a conjecture of Isola and Politi (1990), and extends a similar result for the dimension of invariant subsets of the circle.

Keywords

entropy unimodal maps quadratic polynomials Hölder exponent 

MSC(2010)

37B40 37F20 

Notes

Acknowledgements

This paper is dedicated to the memory of Lei Tan. It would be hard for the author to overstate the support and encouragement got from her, and he will be always grateful. Many of the ideas they discussed are still to be explored, and the author very much hopes those ideas will be eventually worked out with the help of the mathematical community.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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