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arxiv: 2408.00339 · v1 · submitted 2024-08-01 · 🧮 math.DS

Thick attractors with intermingled basins

Abbas Fakhari , Ale Jan Homburg This is my paper

Pith reviewed 2026-05-23 22:37 UTC · model grok-4.3

classification 🧮 math.DS
keywords metric attractorsintermingled basinsrandom walks along orbitsdynamical systemsthick attractorsbasin intermingling
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The pith

Random walks along orbits produce thick metric attractors with intermingled basins.

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

The paper constructs examples of dynamical systems where a metric attractor has positive Lebesgue measure yet its basin is intermingled with another basin, so that every open set meets both basins. The main technique adds random walks along the orbits of a base dynamical system. This modification preserves a metric attractor of positive measure while forcing the basins to mix completely. A sympathetic reader would care because the method supplies elementary constructions for a phenomenon previously seen only in more intricate examples.

Core claim

We construct various novel and elementary examples of dynamics with metric attractors that have intermingled basins. A main ingredient is the introduction of random walks along orbits of a given dynamical system. We develop theory for it and use it in particular to provide examples of thick metric attractors with intermingled basins.

What carries the argument

Random walks along orbits of a base dynamical system, which intermix the basins while preserving a metric attractor of positive measure.

If this is right

  • The modified systems admit metric attractors of positive Lebesgue measure.
  • The basins of attraction become intermingled.
  • Thick attractors (positive measure) can have completely intermingled basins.
  • The random-walk construction works for several different base dynamical systems.

Where Pith is reading between the lines

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

  • The same orbit-modification idea might generate intermingled basins in other well-studied systems such as interval maps or surface diffeomorphisms.
  • Adding controlled noise along orbits could be a general mechanism that turns ordinary attractors into thick ones with mixed basins.
  • One could check whether the boundary between the two basins has positive dimension or supports an invariant measure.

Load-bearing premise

Random walks can be added to orbits so the new map keeps a metric attractor of positive measure.

What would settle it

An explicit example in the paper where simulation or direct calculation shows some nonempty open set lies entirely inside one basin and misses the other.

Figures

Figures reproduced from arXiv: 2408.00339 by Abbas Fakhari, Ale Jan Homburg.

Figure 1
Figure 1. Figure 1: We consider an iterated function system generated by diffeomor￾phisms f0 and f1 on r0, 1s with graphs as depicted. There is an invariant interval Il “ r0, ls. For the inverse maps f ´1 0 and f ´1 1 , the interval Ir “ rr, 1s is invariant [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The skew product system corresponding to the iterated function system generated by maps f0, f1 in left panel, admits two thick attractors. Tak￾ing a random walk on its orbits, and adding a composition with additional random choice from maps ϕ0, ϕ1 as in the right panel, creates thick metric at￾tractors with intermingled basins. Theorem 4.3. Consider the skew product system H on Σ2 ˆ Ω ˆ Σ2 ˆ r0, 1s. Take ν… view at source ↗
Figure 3
Figure 3. Figure 3: We consider an iterated function system generated by diffeomor￾phisms f0 and f1 on r0, 1s with graphs as depicted. There is an invariant interval rl, rs. For a positive integer K1, take K1 mutually disjoint intervals ˜I1, . . . , ˜IK1 inside T, with subintervals Ii Ă ˜Ii . Let f0, f1 be diffeomorphisms on T that are identity maps outside ˜I1 Y ¨ ¨ ¨ Y ˜IK1 and on ˜Ii have graphs as depicted in [PITH_FULL_… view at source ↗
read the original abstract

We construct various novel and elementary examples of dynamics with metric attractors that have intermingled basins. A main ingredient is the introduction of random walks along orbits of a given dynamical system. We develop theory for it and use it in particular to provide examples of thick metric attractors with intermingled basins.

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 / 2 minor

Summary. The manuscript constructs novel and elementary examples of dynamical systems possessing thick metric attractors whose basins are intermingled. The central technique is the introduction of random walks along orbits of a base map, for which supporting theory is developed and then applied to ensure the attractor retains positive measure while the basins become intermingled.

Significance. If the constructions are correct, the work supplies simple, explicit examples in an area of dynamical systems where such attractors are typically obtained only through more elaborate or non-elementary means. The development of the random-walk theory itself may prove reusable for other questions about basin geometry.

minor comments (2)
  1. Notation for the random-walk perturbation (e.g., the probability measure on the orbit) should be introduced once in a dedicated subsection and then used consistently.
  2. A short remark clarifying the ambient manifold or the precise notion of thickness (Lebesgue measure of the attractor) would help readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of the novelty of the elementary examples, and recommendation to accept. The report accurately captures the role of the random-walk theory as a reusable tool for questions on basin geometry.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central contribution is an explicit construction of dynamical examples with thick metric attractors having intermingled basins, achieved by introducing and developing a new theory of random walks along orbits of a base system. This theory is presented as original and then applied to produce the claimed examples; no step reduces a derived quantity to a fitted parameter, self-citation, or definitional tautology. The derivation chain is therefore additive and self-contained rather than circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background from dynamical systems and ergodic theory plus the new but method-level device of random walks; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Existence of a base dynamical system whose orbits admit the superposition of random walks while preserving metric attractor properties.
    Invoked when the random-walk construction is applied to a given system.

pith-pipeline@v0.9.0 · 5559 in / 1072 out tokens · 48485 ms · 2026-05-23T22:37:57.826752+00:00 · methodology

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Reference graph

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