Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

We consider order preserving \(C^3\) circle maps with a flat piece, irrational rotation number and critical exponents \((\ell_1, \ell_2)\). We detect a change in the geometry of the system. For \((\ell_1, \ell_2) \in [1,2]^2\) the geometry is degenerate and it becomes bounded for \((\ell_1, \ell_2) \in [2,\infty)^2 \setminus \{(2,2)\}\). When the rotation number is of the form \([abab\cdots]\); for some \(a,b\in\mathbb{N}^*\), the geometry is bounded for \((\ell_1, \ell_2)\) belonging above a curve defined on \(]1, +\infty [^2\). As a consequence we estimate the Hausdorff dimension of the non-wandering set \(K_f= \mathcal{S}^1 \setminus \bigcup_{i=0}^\infty f^{-i}(U)\). 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.