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arxiv: 2606.12052 · v1 · pith:QASJUHOOnew · submitted 2026-06-10 · 🧮 math.DS

Dimension of the Feigenbaum Attractor

Mark Pollicott This is my paper

Pith reviewed 2026-06-27 08:01 UTC · model grok-4.3

classification 🧮 math.DS
keywords Feigenbaum attractorperiod-doublingHausdorff dimensionrenormalizationattractor dimensionquadratic mapschaos
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The pith

Accurate estimates of the Feigenbaum function g yield improved bounds on the dimension of the period-doubling attractor.

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

The paper describes an effective method to estimate the dimension of the Feigenbaum attractor by converting existing high-accuracy numerical estimates of the function g into estimates for the dimension. This matters because the attractor is the limiting fractal set that appears in the period-doubling route to chaos. A sympathetic reader would care because direct computation of the dimension has been difficult while refined approximations to g are already available from renormalization studies. The approach uses relations from renormalization theory to perform the conversion without separate heavy analysis of the attractor itself.

Core claim

The authors claim that a conversion procedure exists which takes the highly accurate estimates for the Feigenbaum function g and produces improved estimates for the dimension of the attractor X.

What carries the argument

The conversion procedure that maps estimates of the Feigenbaum function g into dimension estimates for the attractor X.

If this is right

  • Improved numerical values for the dimension follow directly from existing high-precision data on g.
  • The method applies to the specific attractor arising in the period-doubling cascade for unimodal maps.
  • No new direct dynamical analysis of the attractor is required once the g estimates are in hand.

Where Pith is reading between the lines

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

  • The same conversion idea could be tested on other renormalization fixed-point problems to see whether dimension estimates simplify similarly.
  • Independent checks might compare the converted bounds against thermodynamic pressure calculations for the same attractor.
  • If the conversion is tight, it would imply that the dimension is effectively determined by the local scaling properties encoded in g.

Load-bearing premise

The conversion from estimates of g to dim(X) is valid and does not introduce uncontrolled errors or require additional unstated properties of the attractor beyond those already known from prior renormalization theory.

What would settle it

A numerical computation of dim(X) by an independent method, such as box-counting on a long orbit approximation of the attractor, that produces a value outside the range obtained from the g-based conversion.

Figures

Figures reproduced from arXiv: 2606.12052 by Mark Pollicott.

Figure 1
Figure 1. Figure 1: (i) A plot of the unimodal g which is a fixed point for the Feigenbaum-Cvitanovic functional equation (1); (ii) A plot of g ◦ g, the middle portion of which looks like an inverted and rescaled plot of g. computed to high precision (e.g., for m ≤ 170, say, the am are computed to in excess of 150 decimal places in [17]). 1.1. Numerical invariants. Associated to the solution R(g) = g are a number of interesti… view at source ↗
read the original abstract

In this note we propose an effective method to estimate the dimension of the Feigenbaum attractor for the period doubling phenomenon. In particular, we will describe a way to convert the highly accurate estimates for $g$ into better estimates on $\dim(X)$.

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

1 major / 0 minor

Summary. The manuscript proposes an effective method to estimate the dimension of the Feigenbaum attractor in the period-doubling cascade by converting highly accurate numerical estimates of the renormalization fixed-point function g into improved estimates for dim(X).

Significance. If a valid, explicitly derived conversion existed that demonstrably improves dimension estimates while inheriting only known properties of the Feigenbaum fixed point, the note could supply a practical numerical tool. No such derivation, formula, error bound, or comparison to independent dimension computations is supplied, so significance cannot be evaluated.

major comments (1)
  1. [Abstract / entire manuscript] The manuscript supplies no derivation, explicit formula, or remainder estimate for the claimed conversion from g-estimates to dim(X). This is load-bearing for the central claim, as the abstract asserts the conversion yields strictly better estimates without introducing uncontrolled errors.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. The central criticism concerns the absence of an explicit derivation, formula, and error estimate for the conversion from g to dim(X). We address this point directly below.

read point-by-point responses

  1. Referee: [Abstract / entire manuscript] The manuscript supplies no derivation, explicit formula, or remainder estimate for the claimed conversion from g-estimates to dim(X). This is load-bearing for the central claim, as the abstract asserts the conversion yields strictly better estimates without introducing uncontrolled errors.

    Authors: We agree that the submitted note does not contain a derivation, explicit formula, or remainder estimate. The manuscript was written as a short announcement of the numerical approach, but the referee is correct that this omission leaves the central claim unsubstantiated. In a revised version we will supply the missing derivation together with an explicit conversion formula and a rigorous remainder bound that shows the error is controlled by the known approximation properties of g. revision: yes

Circularity Check

0 steps flagged

No circularity; no derivation chain or equations supplied

full rationale

The abstract proposes converting accurate g estimates into improved dim(X) bounds for the Feigenbaum attractor but contains no equations, no explicit conversion formula, and no self-citations. Without any load-bearing steps that can be quoted and shown to reduce by construction to fitted inputs or prior self-citations, no circularity is diagnosable. The note is self-contained in its claim of proposing a method but provides no internal derivation to inspect.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are stated or can be inferred.

pith-pipeline@v0.9.1-grok · 5542 in / 923 out tokens · 14755 ms · 2026-06-27T08:01:49.202636+00:00 · methodology

discussion (0)

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Reference graph

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