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# Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

2020, Vol. 16, no. 4, pp.  651-672

Author(s): Ndawa Tangue B.

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
Citation: Ndawa Tangue B., Cherry Maps with Different Critical Exponents: Bifurcation of Geometry, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp.  651-672
DOI:10.20537/nd200409