Arithmetic Geometric Model for the Renormalisation of Bi-critical Irrationally Indifferent Attractors

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points of holomorphic maps with two critical points. The model incorporates arithmetic properties of the rotation number at the fixed point, as well as the “angle” between the two critical points. Using this model for the renormalisation, we build a topological model for the local dynamics of such maps. We also explain the topology of the maximal invariant set for the model, and the dynamics of the map on the maximal invariant set.

💡 Research Summary

The paper develops a comprehensive arithmetic‑geometric framework for studying the local dynamics of holomorphic maps that have a fixed point at the origin with two critical points (a bi‑critical setting). The fixed point is irrationally indifferent, meaning the derivative at zero is a rotation e^{2πiα} with an irrational rotation number α. In the classical uni‑critical case (a single critical point), the arithmetic nature of α (whether it satisfies Brjuno or Herman conditions) determines the existence and size of a Siegel disk and the topology of the maximal invariant set. However, when two critical points are present, an additional geometric parameter appears: the internal angle β between the two critical points (or between the external rays landing at them). This paper introduces a model that simultaneously incorporates both α and β.

The authors first formalize the notion of a bi‑critical irrationally indifferent attractor: two recurrent critical points c₁ and c₂ whose ω‑limit sets coincide and contain the boundary of the Siegel disk (or reduce to the Cremer point). They then construct a renormalisation operator that acts on a pair (α,β). The arithmetic ingredient is the function Q_α(x)=1/(1+min{x,|α|^{-1}−x}), which encodes the continued‑fraction expansion of α and controls the radial scaling of the model. The geometric ingredient is a new change‑of‑coordinates map G_{r,s}, depending on parameters r=|α| and s related to β. G_{r,s} is built from the auxiliary function g_r(w)=|e^{-3πr}−e^{-2πriw}| and enjoys injectivity, periodicity, and a uniform Lipschitz bound.

Using G_{r,s}, the authors define a compact star‑like set M