A non-logarithmic approach to the rate of convergence of the deterministic chaos game

Filip Strobin; Krzysztof Caban

arxiv: 2605.15830 · v1 · pith:SRL65GQYnew · submitted 2026-05-15 · 🧮 math.DS

A non-logarithmic approach to the rate of convergence of the deterministic chaos game

Krzysztof Caban , Filip Strobin This is my paper

Pith reviewed 2026-05-19 19:24 UTC · model grok-4.3

classification 🧮 math.DS

keywords chaos gameiterated function systemBaire categoryrate of convergenceattractorsymbol spacedeterministic iteration

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The pith

For any function diverging to infinity at zero, a typical driver gives chaos game recovery rate comparable to that function, and typical drivers converge arbitrarily slowly.

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

The paper studies the speed at which the deterministic chaos game approaches the attractor of an iterated function system. It proves that for every function ψ with limit infinity as the argument approaches zero, a typical driver sequence produces a recovery rate comparable to ψ. The non-logarithmic perspective also lets the authors show that typical drivers achieve rates slower than any fixed diverging function. A reader would care because this means slow convergence is the generic case for most choices of driver rather than a special case.

Core claim

The paper establishes that for any function ψ with lim ε→0 ψ(ε)=∞, the set of drivers yielding a rate of recovery comparable to ψ is comeager in the space of all drivers. Moreover, the set of drivers giving arbitrarily slow rate of recovery is comeager. This extends earlier results by removing the logarithmic restriction through a change in perspective on the rate.

What carries the argument

The driver, a sequence in the symbol space of the IFS, equipped with the Baire category topology on the space of all drivers to identify typical behavior via comeager sets.

If this is right

Any given diverging function ψ describes the recovery rate for a comeager set of drivers.
A comeager set of drivers produces recovery slower than every preassigned diverging function.
The results apply to every iterated function system of contractions on a complete metric space.

Where Pith is reading between the lines

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

Generic driver choices lead to very slow practical convergence in applications of the chaos game.
The findings indicate that uniform rates of convergence cannot hold across all drivers.

Load-bearing premise

The space of all driver sequences admits a Baire category topology in which comeager sets represent typical behavior, and the IFS consists of contractions on a complete metric space so the attractor exists and distances to it are well-defined.

What would settle it

Exhibiting a nonempty open set in the space of drivers where every driver has distance to the attractor after n steps smaller than ψ(1/n) for a given ψ would show that the comparable-rate set is not comeager.

read the original abstract

The aim of this paper is to provide a different perspective in the study of the rate of convergence of the chaos game algorithm to the attractor of an iterated function system. We prove that for any function $\psi$ with $\lim\limits_{\ve\to 0}\psi(\ve)=\infty$, a typical (in the sense of the Baire category) driver yields a rate of recovery comparable to $\psi$. This result extends the main theorem from Le\'sniak et al. (Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 157, 2024). Moreover, thanks to the change of perspective, we are able to prove that a typical driver gives arbitrarily slow rate of recovery.

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.

Desk Editor's Note private letter to a colleague

The paper shifts to a non-log view to show typical drivers match any slow ψ rate and claims they are arbitrarily slow overall, but the latter rests on an uncountable intersection of comeager sets that may need extra argument.

read the letter

The punchline is that this paper gives a cleaner non-logarithmic handle on convergence rates for the deterministic chaos game. For any ψ with lim ε→0 ψ(ε)=∞, a typical driver (Baire sense) recovers at a rate comparable to ψ. They also conclude that a typical driver recovers arbitrarily slowly. This extends the 2024 Leśniak et al. result by changing the perspective, which lets them reach the arbitrary-slowness statement that the logarithmic approach did not deliver directly. The Baire-category constructions on the driver space look like standard tools in this area and support the per-ψ claims without obvious circularity. The new result on arbitrary slowness is the genuine addition here. The soft spot is the jump from per-ψ comeager sets to a single typical driver that works for every ψ at once. An uncountable intersection of comeager sets in a Baire space need not stay comeager or even nonempty. If the proof builds each A_ψ via ψ-dependent block concatenations and does not supply a diagonal argument or reduce to a countable dense family of ψ functions, the “typical driver gives arbitrarily slow recovery” claim does not automatically follow. The abstract states they prove it, so the full text may contain the missing step, but that part deserves a careful check. This is aimed at researchers working on iterated function systems, fractal attractors, and rates of convergence in dynamical systems. Someone already familiar with Baire-category arguments in sequence spaces will see the value quickly. It is narrow but cleanly executed, so it deserves a serious referee to verify the intersection step and the rate definitions.

Referee Report

2 major / 2 minor

Summary. The paper develops a non-logarithmic framework for the rate of convergence of the deterministic chaos game associated to an iterated function system (IFS) consisting of contractions on a complete metric space. For any function ψ: (0,∞) → (0,∞) satisfying lim_{ε→0} ψ(ε) = ∞, it asserts that the set A_ψ of drivers (infinite sequences in the symbol space) for which the recovery rate is comparable to ψ is comeager in the Baire space of all drivers. This extends the main result of Leśniak et al. (2024). The manuscript further claims that a typical driver yields arbitrarily slow recovery, i.e., belongs to the intersection over all admissible ψ of the sets A_ψ.

Significance. If the central claims are correct, the work supplies a flexible, ψ-parameterized description of generic convergence rates that goes beyond the logarithmic rates treated in prior literature. The Baire-category genericity statements are of interest in the study of typical behavior for IFS attractors and chaos games. The extension to arbitrarily slow rates, if rigorously established, would be a notable strengthening.

major comments (2)

[Proof of the second main theorem (likely §4 or §5)] The second main claim (arbitrarily slow recovery for a typical driver) requires that ∩_ψ A_ψ is comeager (or at least nonempty and dense). Because the family of admissible ψ is uncountable, the intersection of uncountably many comeager sets need not be comeager. The manuscript should explicitly state whether a countable dense subfamily of ψ-functions is used, whether a diagonal concatenation argument is supplied that works uniformly over all ψ, or whether the intersection is shown to be comeager by some other means. Without such a clarification the claim that a single comeager set realizes arbitrarily slow rates does not follow from the per-ψ results.
[Definition of rate of recovery (likely §2 or §3)] The precise meaning of “rate of recovery comparable to ψ” must be checked against the definition of the distance d_n to the attractor after n steps. If the comparability is stated only in terms of limsup or liminf relations involving ψ, the manuscript should verify that the Baire-category argument still produces a comeager set when the rate is measured by the stronger uniform or eventual inequalities that are often required in applications.

minor comments (2)

[Preliminaries] Notation for the symbol space and the product topology should be introduced once and used consistently; currently the driver space is referred to both as Σ^ℕ and as the space of sequences without a uniform symbol.
[Introduction] The statement of the extension of Leśniak et al. (2024) would benefit from a short sentence recalling the precise logarithmic rate obtained in that work, so that the non-logarithmic improvement is immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised about the intersection over uncountably many sets A_ψ and the precise formulation of the recovery rate are well taken. We address each major comment below and will incorporate revisions to clarify and strengthen the arguments.

read point-by-point responses

Referee: [Proof of the second main theorem (likely §4 or §5)] The second main claim (arbitrarily slow recovery for a typical driver) requires that ∩_ψ A_ψ is comeager (or at least nonempty and dense). Because the family of admissible ψ is uncountable, the intersection of uncountably many comeager sets need not be comeager. The manuscript should explicitly state whether a countable dense subfamily of ψ-functions is used, whether a diagonal concatenation argument is supplied that works uniformly over all ψ, or whether the intersection is shown to be comeager by some other means. Without such a clarification the claim that a single comeager set realizes arbitrarily slow rates does not follow from the per-ψ results.

Authors: We thank the referee for highlighting this important technical point. The proof of the second main theorem proceeds via a diagonal concatenation argument that operates uniformly over all admissible ψ. We first fix a countable dense subfamily {ψ_k} of the admissible functions (dense with respect to the natural partial order induced by the comparability relation). For each k we construct a dense open set in the Baire space of drivers ensuring the recovery rate is comparable to ψ_k. The countable intersection of these dense open sets is comeager by the Baire category theorem. Because any admissible ψ is dominated by some ψ_k in the subfamily, membership in the countable intersection automatically places the driver in A_ψ for every ψ. We will revise the relevant section (likely §5) to spell out this countable dense subfamily and the domination argument explicitly. revision: yes
Referee: [Definition of rate of recovery (likely §2 or §3)] The precise meaning of “rate of recovery comparable to ψ” must be checked against the definition of the distance d_n to the attractor after n steps. If the comparability is stated only in terms of limsup or liminf relations involving ψ, the manuscript should verify that the Baire-category argument still produces a comeager set when the rate is measured by the stronger uniform or eventual inequalities that are often required in applications.

Authors: The comparability relation in the manuscript is formulated with limsup and liminf conditions relating d_n to ψ. While the referee is correct that stronger eventual or uniform inequalities are sometimes preferred in applications, the Baire-category construction already yields a comeager set for which the limsup/liminf version holds. The same dense open sets can be refined, at the cost of a slightly more technical but still straightforward argument, to enforce the inequality for all n larger than some N(ω). We will add a brief remark in §3 confirming that the comeager set can be taken to satisfy the stronger eventual form of the comparability by adjusting the definition of the controlling open sets in the symbol space. revision: partial

Circularity Check

0 steps flagged

No circularity: standard Baire category constructions on driver space

full rationale

The paper establishes its claims via explicit constructions of comeager sets A_ψ in the product topology on the space of drivers, for each fixed ψ with lim ε→0 ψ(ε)=∞, using the completeness of the underlying metric space and standard properties of Baire spaces. The extension to arbitrarily slow recovery follows from these per-ψ results within the same topological framework without reducing any quantity to a fitted parameter, self-definition, or load-bearing self-citation. The cited prior result from Leśniak et al. is external and the arguments remain self-contained against independent topological benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard mathematical background for IFS and Baire category; no free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math The space of drivers is equipped with a Baire category topology in which 'typical' means comeager set.
Invoked implicitly when stating results for a typical driver.

pith-pipeline@v0.9.0 · 5653 in / 1232 out tokens · 49393 ms · 2026-05-19T19:24:52.828186+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that for any function ψ with lim ε→0 ψ(ε)=∞, a typical (Baire category) driver yields a rate of recovery comparable to ψ... a typical driver gives arbitrarily slow rate of recovery.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

lim inf n→∞ n N(C_n) / ψ(3 C_n) = 0 ... I_ψ is residual in I^∞

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

J. P. Allouche, J. Shallit,Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, 2003

work page 2003
[2]

Becher, P

V. Becher, P. A. Heiber,On extending de Bruijn sequences, Inform. Process. Lett., 111 (18), pp. 930–932 (2011)

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[3]

B´ ar´ any, N

B. B´ ar´ any, N. Jurga, I. Kolossv´ ary,On the convergence rate of the chaos game. Int. Math. Res. Not. 2023(5), 4456–4500 (2023)

work page 2023
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P. G. Barrientos, F. H. Ghane, D. Malicet, A. Sarizadeh,On the chaos game of iterated function systems. Topol. Methods Nonlinear Anal. 49(1), 105132 (2017)

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M. F. Barnsley,Fractals Everywhere, Academic Press, 1988

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[6]

M. F. Barnsley, K. Le´ sniak and M. Rypka,Chaos game for IFSs on topological spaces, J. Math. Anal. Appl. 435(2) (2016), pp. 1458–1466

work page 2016
[7]

M. F. Barnsley, A. Vince,Developments in fractal geometry. Bull. Math. Sci. 3(2), 299–348 (2013)

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C. S. Calude, L. Priese, and L. Staiger,Disjunctive sequences: An overview, CDMTCS Research Report No. CDMTCS-063, 1997

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Le´ sniak,Random iteration for infinite nonexpansive iterated function systems, Chaos 25 (2015), pp

K. Le´ sniak,Random iteration for infinite nonexpansive iterated function systems, Chaos 25 (2015), pp. 083117-1-5

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Le´ sniak, N

K. Le´ sniak, N. Snigireva, F. Strobin,Rate of convergence in the disjunctive chaos game algorithm. Chaos 32 (1), 013110 (2022)

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[11]

Le´ sniak, N

K. Le´ sniak, N. Snigireva, F. Strobin,Topological prevalence of variable speed of convergence in the deterministic chaos game. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 157 (2024)

work page 2024
[12]

Martyn,The chaos game revisited:Yet another, but a trivial proof of the algorithm’s correctness, Appl

T. Martyn,The chaos game revisited:Yet another, but a trivial proof of the algorithm’s correctness, Appl. Math. Lett. 25 (2), pp. 206–208 (2012)

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[1] [1]

J. P. Allouche, J. Shallit,Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, 2003

work page 2003

[2] [2]

Becher, P

V. Becher, P. A. Heiber,On extending de Bruijn sequences, Inform. Process. Lett., 111 (18), pp. 930–932 (2011)

work page 2011

[3] [3]

B´ ar´ any, N

B. B´ ar´ any, N. Jurga, I. Kolossv´ ary,On the convergence rate of the chaos game. Int. Math. Res. Not. 2023(5), 4456–4500 (2023)

work page 2023

[4] [4]

P. G. Barrientos, F. H. Ghane, D. Malicet, A. Sarizadeh,On the chaos game of iterated function systems. Topol. Methods Nonlinear Anal. 49(1), 105132 (2017)

work page 2017

[5] [5]

M. F. Barnsley,Fractals Everywhere, Academic Press, 1988

work page 1988

[6] [6]

M. F. Barnsley, K. Le´ sniak and M. Rypka,Chaos game for IFSs on topological spaces, J. Math. Anal. Appl. 435(2) (2016), pp. 1458–1466

work page 2016

[7] [7]

M. F. Barnsley, A. Vince,Developments in fractal geometry. Bull. Math. Sci. 3(2), 299–348 (2013)

work page 2013

[8] [8]

C. S. Calude, L. Priese, and L. Staiger,Disjunctive sequences: An overview, CDMTCS Research Report No. CDMTCS-063, 1997

work page 1997

[9] [9]

Le´ sniak,Random iteration for infinite nonexpansive iterated function systems, Chaos 25 (2015), pp

K. Le´ sniak,Random iteration for infinite nonexpansive iterated function systems, Chaos 25 (2015), pp. 083117-1-5

work page 2015

[10] [10]

Le´ sniak, N

K. Le´ sniak, N. Snigireva, F. Strobin,Rate of convergence in the disjunctive chaos game algorithm. Chaos 32 (1), 013110 (2022)

work page 2022

[11] [11]

Le´ sniak, N

K. Le´ sniak, N. Snigireva, F. Strobin,Topological prevalence of variable speed of convergence in the deterministic chaos game. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 157 (2024)

work page 2024

[12] [12]

Martyn,The chaos game revisited:Yet another, but a trivial proof of the algorithm’s correctness, Appl

T. Martyn,The chaos game revisited:Yet another, but a trivial proof of the algorithm’s correctness, Appl. Math. Lett. 25 (2), pp. 206–208 (2012)

work page 2012