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.