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 Rus. J. Nonlin. Dyn., 2020, Volume 16, Number 4, Pages 651–672 (Mi nd735)

Mathematical problems of nonlinearity

Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

B. Ndawa Tangue

Institute of Mathematics and Physical Sciences Avakpa, Porto-Novo, 613 Benin

Abstract: We consider order-preserving $C^3$ circle maps with a flat piece, irrational rotation number and critical exponents $(l_1, l_2)$.
We detect a change in the geometry of the system. For $(l_1, l_2) \in [1, 2]^2$ the geometry is degenerate and becomes bounded for $(l_1, l_2) \in [2, \infty)^2 \backslash \{(2, 2)\}$. When the rotation number is of the form $[abab \ldots]$; for some $a, b \in \mathbb{N}^*$, the geometry is bounded for $(l_1, l_2)$ belonging above a curve defined on $]1, +\infty[^2$. As a consequence, we estimate the Hausdorff dimension of the nonwandering set $K_f=\mathcal{S}^1 \backslash \bigcup^\infty_{i=0}f^{-i}(U)$. Precisely, the Hausdorff dimension of this set is equal to zero when the geometry is degenerate and it is strictly positive when the geometry is bounded.

Keywords: circle map, irrational rotation number, flat piece, critical exponent, geometry, Hausdorff dimension

 Funding Agency The author was partially supported by the Centre d’Excellence Africain en Science Mathématiques et Applications (CEA-SMA).

DOI: https://doi.org/10.20537/nd200409

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MSC: 37E10
Accepted:27.10.2020

Citation: B. Ndawa Tangue, “Cherry Maps with Different Critical Exponents: Bifurcation of Geometry”, Rus. J. Nonlin. Dyn., 16:4 (2020), 651–672

Citation in format AMSBIB
\Bibitem{Nda20} \by B.~Ndawa Tangue \paper Cherry Maps with Different Critical Exponents: Bifurcation of Geometry \jour Rus. J. Nonlin. Dyn. \yr 2020 \vol 16 \issue 4 \pages 651--672 \mathnet{http://mi.mathnet.ru/nd735} \crossref{https://doi.org/10.20537/nd200409} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4198786}