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arxiv: 2606.04783 · v1 · pith:QOHJYXDDnew · submitted 2026-06-03 · 🧮 math.DS

Intermingled basins: Kan's example on the Riemann sphere

Abbas Fakhari , Ale Jan Homburg , Sebastian van Strien This is my paper

Pith reviewed 2026-06-28 03:44 UTC · model grok-4.3

classification 🧮 math.DS
keywords intermingled basinsKan's exampleRiemann sphereskew productsholomorphic mapsattractorsexpanding circle mapsdynamical systems
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The pith

Skew product systems of holomorphic maps on the Riemann sphere have three attractors with intermingled basins of full support.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Kan's construction of systems with two intermingled attractors to the Riemann sphere. Kan used skew products of interval diffeomorphisms forced by expanding circle maps; here those diffeomorphisms are polynomials, allowing holomorphic extension to the sphere. The authors prove that the lifted systems possess three attractors rather than two. A sympathetic reader would care because the result shows intermingled basins persist when moving from real one-dimensional maps to complex dynamics on the sphere.

Core claim

We consider the resulting skew product systems of holomorphic maps on the Riemann sphere forced by expanding circle maps, and establish the existence of three attractors whose basins of attraction all have full support and are thus intermingled.

What carries the argument

Skew product systems of holomorphic maps on the Riemann sphere forced by expanding circle maps, obtained by lifting polynomial interval diffeomorphisms.

If this is right

  • The construction produces at least three coexisting attractors, each with a basin of full support.
  • The basins remain intermingled, so each is dense wherever the others are dense.
  • Kan's original phenomenon on the interval lifts directly to the complex sphere setting.
  • The number of attractors with this property increases from two to three under the holomorphic extension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar lifts might allow constructions with four or more intermingled attractors by choosing different polynomial degrees.
  • The result suggests that basin intermingling is robust under complexification of real skew products.
  • Numerical iteration of random points on the sphere could be used to estimate the relative sizes of the three basins in concrete examples.

Load-bearing premise

The interval diffeomorphisms are polynomial maps and can therefore be extended to holomorphic maps on the Riemann sphere.

What would settle it

A dense open set of points on the Riemann sphere whose forward orbits under the skew product fail to approach any of the three claimed attractors would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.04783 by Abbas Fakhari, Ale Jan Homburg, Sebastian van Strien.

Figure 1
Figure 1. Figure 1: Pictures of filled Julia sets for α “ 0.3 and L “ 3. The sets W s 0 p1q (light blue region in the left picture) and W s 1{2 p0q (dark blue region in the right picture) are bounded by the Julia sets J0 (encircling 1, containing 0) and J1{2 (encircling 0, containing 1) respectively. 6 0 -6 -8 -4 0 4 6 0 -6 -4 -2 0 2 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerically calculated sets Ws u p0q, Ws u p1q for two irrational values of u, with α “ 0.3, L “ 3. For the left picture we chose u “ 1{ ? 3, and u “ 1{ ? 5 for the right picture. Colors are as in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Iterates of an open set O that intersect the basin of attraction of T ˆ t0u contain parts that accumulate onto T ˆ t0u. To prove that the basins of T ˆ t0u and T ˆ t1u are intermingled, we show that further iterates contain parts that accumulate onto T ˆ t1u. Proof of Bs p0q Ă Bs p1q. Take a point in Bs p0q and a neighborhood O of it. Then O intersects Ws p0q in a set of positive measure. We must show that… view at source ↗
read the original abstract

Kan's discovery of dynamical systems with two attractors whose basins of attraction both have full support, featured specific examples of skew product systems of interval diffeomorphisms forced by expanding circle maps. The interval diffeomorphisms are polynomial maps and can hence be considered on the Riemann sphere. We consider the resulting skew product systems of holomorphic maps on the Riemann sphere forced by expanding circle maps, and establish the existence of three attractors whose basins of attraction all have full support and are thus intermingled.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript extends Kan's skew-product construction of systems with intermingled basins from interval diffeomorphisms to holomorphic maps on the Riemann sphere. Polynomial interval maps are lifted to the sphere and forced by expanding circle maps; the authors prove the existence of three attractors whose basins each have full support (hence are intermingled).

Significance. If the result holds, the paper supplies a holomorphic analogue of Kan's example with one additional attractor. This is a natural and potentially useful contribution to the literature on basin intermingling in complex dynamics, obtained by a standard extension of polynomial maps to CP^1.

minor comments (1)
  1. [§1] The abstract states that the interval diffeomorphisms are polynomial and hence extend to the Riemann sphere; a one-sentence reminder in §1 of the precise degree or leading-term condition used for the extension would aid readers unfamiliar with the real-to-complex transition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. The report correctly identifies the main contribution as a holomorphic analogue of Kan's skew-product construction on the Riemann sphere, yielding three intermingled attractors.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained extension of external reference

full rationale

The paper extends Kan's skew-product construction (external citation) to holomorphic maps on the Riemann sphere via polynomial interval diffeomorphisms, proving existence of three attractors with full-support intermingled basins. No load-bearing step reduces by definition, fitted parameter, or self-citation chain to the target claim; the polynomial-to-Riemann-sphere step is a routine fact in complex dynamics, and the existence result is presented as a direct mathematical argument without renaming, ansatz smuggling, or uniqueness imported from the authors' prior work. The derivation chain is independent of the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract supplies only one explicit modeling choice; no free parameters, invented entities, or additional axioms are mentioned.

axioms (1)
  • domain assumption Interval diffeomorphisms are polynomial maps and can therefore be considered on the Riemann sphere
    This premise is invoked to justify lifting the original real maps to holomorphic maps on the sphere.

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