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The infimum of Lipschitz constants in the conjugacy class of an interval map


Authors: Jozef Bobok and Samuel Roth
Journal: Proc. Amer. Math. Soc. 147 (2019), 255-269
MSC (2010): Primary 37E05; Secondary 26A16, 37B40
DOI: https://doi.org/10.1090/proc/14255
Published electronically: October 18, 2018
MathSciNet review: 3876747
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Abstract: How can we interpret the infimum of Lipschitz constants in the conjugacy class of an interval map? For a positive entropy map $f$, the exponential $\exp h(f)$ of the topological entropy gives a well-known lower bound. In the case of a countably piecewise monotone map that is topologically mixing and Markov, we characterize the infimum $\Lambda (f)$ of Lipschitz constants as the exponential of the Salama entropy of a certain reverse Markov chain associated with the map. Dynamically, this number represents the exponential growth rate of the number of iterated preimages of nearly any point; we show that it can be strictly larger than $\exp h(f)$. In addition we prove that if $f$ is piecewise monotone or $C^{\infty }$, these two quantities $\Lambda (f)$ and $\exp h(f)$ are equal.


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Additional Information

Jozef Bobok
Affiliation: Czech Technical University in Prague, FCE, Thákurova 7, 166 29 Praha 6, Czech Republic
MR Author ID: 305416
Email: jozef.bobok@cvut.cz

Samuel Roth
Affiliation: Silesian University in Opava, Na Rybničku 626/1, 746 01 Opava, Czech Republic
MR Author ID: 1155238
Email: samuel.roth@math.slu.cz

Keywords: Interval map, Lipschitz constant, topological entropy, countable Markov shift
Received by editor(s): April 18, 2017
Received by editor(s) in revised form: March 12, 2018, and April 17, 2018
Published electronically: October 18, 2018
Additional Notes: The first author was supported by the European Regional Development Fund, project No. CZ.02.1.01/0.0/0.0/16_019/0000778.
The second author was supported by RVO funding for IČ47813059
Communicated by: Nimish Shah
Article copyright: © Copyright 2018 American Mathematical Society