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Rigidity of Fibonacci Circle Maps with a Flat Piece and Different Critical Exponents

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Abstract

We consider order preserving \(C^3\) circle maps with a flat piece, Fibonacci rotation number, and negative Schwarzian derivative where the critical exponents (the degrees of the singularities at the boundary of the flat piece) might be different.This paper treats the rigidity (geometrical) characteristic of a map of our class. We prove that when the critical exponents belong to \((1,2)^2\), the geometry of the system is degenerate (double exponentially fast). As a consequence, the renormalization diverges, and the rigidity (geometric) class depends on three ordered pairs.

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Notes

  1. $$\begin{aligned} Dim_HK_f:=\inf _{s\ge 0}\left\{ H_s(K_f):=\inf _{\delta >0}\left\{ \sum |U_i|^{s},\; K_f\subset \cup U_i,\;\;|U_i|<\delta \right\} =0\right\} . \end{aligned}$$

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Acknowledgements

The author would like to thank the referee whose valuable comments helped to improve the exposition of the paper. I sincerely thank Prof. M. Martens and Prof. Dr. L. Palmisano for introducing me (at Institute of Mathematics and Physical Sciences (IMSP)) to the subject of this paper, for valuable advice, continuous encouragement and helpful discussions. Part of the research for this paper took place at National Advanced School of Engineering, Yaounde (ENSPY) and University of Dauala (UD). The author would like to thank the IMSP, ENSPY, UD, and in particular Prof. Ferdinand NGAKEU, Prof. Joel TOSSA, Prof. Carlos OGOUYANDJOU, Prof. Léonard TODJIHOUNDE, Prof. Thomas BOUETOU BOUETOU, Prof. Bridinette THIODJIO SENDJA, épse FAN DIO, Prof. Louis Aimé FONO and Prof. Anatole TEMGOUA for continuous encouragement or their hospitality.

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  1. Computer Engineering, University Institute of Technology, University of Ngaoundere, N1, Ngaoundere, 455, Cameroon

    Bertuel TANGUE NDAWA

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NDAWA, B.T. Rigidity of Fibonacci Circle Maps with a Flat Piece and Different Critical Exponents. J Dyn Control Syst (2024). https://doi.org/10.1007/s10883-024-09687-z

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