Spectral properties of the period-doubling operator

TL;DR

This paper investigates the spectral properties of the Feigenbaum period-doubling operator in a space of bounded analytical functions, revealing a higher-dimensional unstable manifold than previously conjectured.

Contribution

It provides a detailed spectral analysis of the operator in a new function space, challenging prior assumptions about the dimension of the unstable manifold.

Findings

The spectrum differs from that in the space of even analytical functions.

The unstable manifold dimension is three, not one as previously conjectured.

Comparison of different approaches and algorithms for spectral analysis.

Abstract

We compute the spectrum of the Feigenbaum period-doubling operator in the space of bounded analytical functions in an ellipse. The spectral properties of the period-doubling operator in this space are not the same as in the space of even analytical functions. In particular, it was found that the dimension of the unstable manifold is not one (Feigenbaum's conjecture), but three. We analyze several articles devoted to this problem and compare different approaches and algorithms.