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 $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$. We detect a change in the geometry of the system. For $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)\in [1,2{]}^2$ the geometry is degenerate and it becomes bounded for $(\ell_1, \ell_2) \in [2,\infty)^2 \setminus {(2,2)}$\mathrm{.}Whentherotationnumberisoftheform$[abab\cdots]$; for some $a,b\in {\mathbb{N}}^{\ast }$, the geometry is bounded for $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$ belonging above a curve defined on $]1,+\mathrm{\infty }{[}^2$. 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.