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

TL;DR

This paper investigates the geometric properties of Fibonacci circle maps with a flat piece and critical exponents, revealing degeneracy in system geometry and divergence in renormalization for certain parameter ranges.

Contribution

It characterizes the geometry of Fibonacci circle maps with flat pieces and critical exponents in (1,2), showing degeneracy and dependence of rigidity classes on specific parameters.

Findings

Geometry is degenerate for (l1, l2) in (1,2)^2.

Renormalization diverges in this parameter range.

Rigidity classes depend on three pairs of parameters.

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}