Geodesic
Cherry Maps with Different Critical Exponents: Bifurcation of Geometry
Russian journal of nonlinear dynamics, Tome 16 (2020) no. 4, pp. 651-672

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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
Mots-clés : Hausdorff dimension. 
@article{ND_2020_16_4_a8,
     author = {B. Ndawa Tangue},
     title = {Cherry {Maps} with {Different} {Critical} {Exponents:} {Bifurcation} of {Geometry}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {651--672},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2020_16_4_a8/}
}
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B. Ndawa Tangue. Cherry Maps with Different Critical Exponents: Bifurcation of Geometry. Russian journal of nonlinear dynamics, Tome 16 (2020) no. 4, pp. 651-672. http://geodesic.mathdoc.fr/item/ND_2020_16_4_a8/