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 $(c_u(f),c_u^{\prime }(f))$, $(c_{+}(f),c_{+}^{\prime }(f))$ and $(c_s(f),c_s^{\prime }(f))$.\vspace{0.5cm}

arXiv comment42 pages