Rigidity of Fibonacci Circle Maps with a Flat Piece and Different Critical Exponents

Abstract

We consider order preserving $C^3$ circle maps with a flat piece, Fibonacci rotation number, critical exponents $(\ell_1, \ell_2)$ and negative shwarzian derivative. This paper treat the geometry characteristic of the non-wondering (cantor (fractal)) set from a map of our class. We prove that, for $(\ell_1, \ell_2)$ in $(1,2)^2$, the geometry of system is degenerate (double exponentially fast). As consequences, the renormalization diverges and the geometric (rigidity) class depends on the three couples $(c_u(f), c'_u(f) )$, $( c_+(f), c'_+(f))$ and $(c_s(f), c'_s(f) )$.\vspace{0.5cm}