The local Hölder exponent for the entropy of real unimodal maps
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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.
Keywordsentropy unimodal maps quadratic polynomials Hölder exponent
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.
- 1.Bandtlow O F, Rugh H H. Entropy-continuity for interval maps with holes. Ergodic Theory Dynam Systems, 2018, in pressGoogle Scholar
- 4.Dobbs N, Mihalache N. Diabolical entropy. ArXiv:1610.02563, 2016Google Scholar
- 5.Douady A. Topological entropy of unimodal maps: Monotonicity for quadratic polynomials. In: Real and Complex Dynamical Systems, vol. 464. Dordrecht: Kluwer, 1995, 65–87Google Scholar
- 6.Dudko D, Schleicher D. Core entropy of quadratic polynomials. ArXiv:1412.8760v1, 2014Google Scholar
- 8.Guckenheimer J. The growth of topological entropy for one dimensional maps. In: Global Theory of Dynamical Systems. Lecture Notes in Mathematics, vol. 819. Berlin: Springer, 1980, 216–223Google Scholar
- 10.Jung W. Core entropy and biaccessibility of quadratic polynomials. ArXiv:1401.4792, 2014Google Scholar
- 11.Milnor J, Thurston W. On iterated maps of the interval. In: Dynamical Systems. Lecture Notes in Mathematics, vol. 1342. Berlin: Springer, 1988, 465–563Google Scholar