Common Limits of Fibonacci Circle Maps

TL;DR

This paper investigates the limiting behavior of Fibonacci circle maps as the critical exponent approaches infinity, revealing that the limits are identical for certain classes and exhibit complex dynamics despite non-analyticity.

Contribution

It establishes the existence and equality of limit maps for critical circle homeomorphisms and coverings with golden mean rotation, highlighting their complex dynamics.

Findings

Limits exist for both critical circle homeomorphisms and coverings as the critical exponent tends to infinity.

The limit maps are identical for these classes.

Limit maps are non-analytic at the critical point but display non-trivial complex dynamics.

Abstract

We show that limits for the critical exponent tending to \infty exist in both critical circle homeomorphisms of golden mean rotation number and Fibonacci circle coverings. Moreover, they are the same. The limit map is not analytic at the critical point, which is flat, but has non-trivial complex dynamics.