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.
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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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In Memory of Lei Tan
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Tiozzo, G. The local Hölder exponent for the entropy of real unimodal maps. Sci. China Math. 61, 2299–2310 (2018). https://doi.org/10.1007/s11425-017-9293-7
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DOI: https://doi.org/10.1007/s11425-017-9293-7