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

We consider order preserving C3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>3$$\end{document} circle maps with a flat piece, Fibonacci rotation number, and negative Schwarzian derivative where the critical exponents (the degrees of the singularities at the boundary of the flat piece) might be different.This paper treats the rigidity (geometrical) characteristic of a map of our class. We prove that when the critical exponents belong to (1,2)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1,2)<^>2$$\end{document}, the geometry of the system is degenerate (double exponentially fast). As a consequence, the renormalization diverges, and the rigidity (geometric) class depends on three ordered pairs.