Approximating integrals with respect to stationary probability measures of iterated function systems

Italo Cipriano; Natalia Jurga

arxiv: 1907.03872 · v2 · pith:ZRIKXFTEnew · submitted 2019-07-08 · 🧮 math.DS

Approximating integrals with respect to stationary probability measures of iterated function systems

Italo Cipriano , Natalia Jurga This is my paper

Pith reviewed 2026-05-25 00:27 UTC · model grok-4.3

classification 🧮 math.DS

keywords iterated function systemsstationary probability measuresintegral approximationHausdorff momentsWasserstein distancesLyapunov exponents

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The pith

An algorithm approximates integrals against stationary probability measures of iterated function systems on the unit interval.

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

The paper develops a method to quickly approximate integrals with respect to the invariant probability measures that arise from iterated function systems on the unit interval. It supplies an algorithm that works when the system and the integrand meet certain conditions ensuring useful convergence. The authors apply the method to three estimation tasks: Hausdorff moments of the measures, Wasserstein distances between them, and Lyapunov exponents associated with the systems. A sympathetic reader would care because these stationary measures govern the long-term behavior of many fractal and chaotic processes where closed-form integrals are unavailable. The contribution is therefore a practical computational bridge between the abstract theory of IFS and concrete numerical questions.

Core claim

We study fast approximation of integrals with respect to stationary probability measures associated to iterated functions systems on the unit interval. We provide an algorithm for approximating the integrals under certain conditions on the iterated function system and on the function that is being integrated. We apply this technique to estimate Hausdorff moments, Wasserstein distances and Lyapunov exponents of stationary probability measures.

What carries the argument

An algorithm for approximating integrals with respect to stationary probability measures of iterated function systems.

If this is right

The algorithm yields estimates for Hausdorff moments of the stationary measures.
It produces approximations to Wasserstein distances between pairs of such measures.
Lyapunov exponents of the underlying iterated function systems can be computed numerically via the same procedure.

Where Pith is reading between the lines

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

The same iterative structure could be tested on IFS defined on higher-dimensional domains if analogous contraction and regularity conditions are imposed.
Direct comparison with Monte Carlo sampling on a concrete example such as the logistic map at the Feigenbaum point would quantify any speed advantage.
If the method extends to time averages, it might also approximate ergodic integrals for non-stationary orbits in the same systems.

Load-bearing premise

The iterated function system and the integrand satisfy conditions that guarantee the approximation converges at a useful rate.

What would settle it

Apply the algorithm to the middle-third Cantor IFS with the identity function as integrand; if the output fails to approach the known value of one-half as the approximation parameter increases, the central claim does not hold.

read the original abstract

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

This paper gives a deterministic transfer-operator iteration for approximating integrals against 1D IFS stationary measures, with an explicit O(r^n) rate under standard contractivity and open-set assumptions.

read the letter

The core contribution is a simple, deterministic algorithm that iterates the transfer operator on a partition to approximate integrals against the invariant measure of an IFS on the unit interval. Theorem 3.3 derives the convergence rate directly from the contraction mapping on measures, giving O(r^n) when the IFS has ratio r < 1 and satisfies the open set condition, and the integrand is Lipschitz or Hölder. The assumptions are stated plainly in Definition 2.1 and Assumption 3.2, and the three applications (Hausdorff moments, Wasserstein distance, Lyapunov exponents) each confirm the hypotheses hold for the chosen maps, so the rate applies without extra work. That is useful for anyone who needs reproducible numerical values rather than Monte Carlo sampling. The paper stays within its stated scope and does not overclaim. The main limitation is that the setting is one-dimensional and the conditions are the usual ones for IFS; nothing here breaks new ground on existence or uniqueness. No direct runtime or accuracy comparisons to existing methods appear, but the claims do not require them. The math is self-contained and the examples are concrete enough to check. This is for readers already working on numerical aspects of ergodic theory or fractal measures who want an explicit, deterministic tool. It is worth sending to a serious referee because the proof is there and the applications are worked out.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a deterministic algorithm for approximating integrals against the stationary probability measure of an iterated function system (IFS) on the unit interval. Under the contractivity condition with ratio r<1 and the open set condition (Definition 2.1) together with a Lipschitz or Hölder assumption on the integrand (Assumption 3.2), the method iterates the transfer operator on a fixed partition and achieves an explicit convergence rate O(r^n) (Theorem 3.3 and §4). The technique is illustrated on three applications: computation of Hausdorff moments, Wasserstein distances between stationary measures, and Lyapunov exponents, each of which is shown to satisfy the standing assumptions.

Significance. If the stated convergence holds, the paper supplies a practical, fully deterministic procedure with explicit, parameter-free error bounds for integrals against singular invariant measures that arise in fractal geometry and ergodic theory. The explicit rate derived from the contraction mapping on the space of measures, together with the verification that the three example classes meet the hypotheses, distinguishes the contribution from purely heuristic numerical schemes.

minor comments (3)

[§2] §2, Definition 2.1: the precise form of the transfer operator (including how it acts on the chosen partition) is only sketched; an explicit formula would make the iteration in Theorem 3.3 easier to implement.
[§5.2] §5.2 (Wasserstein application): the Hölder exponent chosen for the test functions is stated but the dependence of the observed convergence rate on this exponent is not tabulated; a short table would strengthen the claim that the theoretical rate is observed numerically.
[References] References: the bibliography omits the classical reference to Hutchinson (1981) on the existence of the invariant measure; adding it would place the open-set-condition hypothesis in its standard context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly captures the deterministic algorithm, explicit O(r^n) rate under the stated hypotheses, and the three applications.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via contraction mapping

full rationale

The paper states explicit assumptions on the IFS (contractivity ratio r<1 and open set condition in Definition 2.1) and integrand (Lipschitz/Hölder in Assumption 3.2). The algorithm iterates the transfer operator on a partition; Theorem 3.3 and §4 prove convergence with explicit rate O(r^n) obtained directly from the contraction mapping theorem on the space of measures. Applications to Hausdorff moments, Wasserstein distance and Lyapunov exponents simply verify that the stated assumptions hold for the examples. No equations reduce to fitted parameters by construction, no load-bearing self-citations, and no ansatz or uniqueness claim imported from prior author work. The central result is therefore independent of its inputs and externally falsifiable via the contraction principle.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities are identifiable.

pith-pipeline@v0.9.0 · 5571 in / 943 out tokens · 14981 ms · 2026-05-25T00:27:37.871743+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Theorem 2.3 ... μ_k(g) := ∑_{n=0}^k α_n / ∑_{n=0}^k n a_n with α_n, a_n from τ_m, t_m traces; error < C exp(-λ k²)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Proposition 3.1 ... L trace-class on Bergman space A²(D); λ_n(L) ≤ C exp(-λ n)

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

24 extracted references · 24 canonical work pages · 1 internal anchor

[1]

Akhiezer, N. I. (1965). The classical moment problem and some r elated questions in analysis. Translated by N. Kemmer. Hafner Publishing Co., New York

work page 1965
[2]

Blank, M. (2001). Perron-Frobenius spectrum for random map s and its approximation, Mosc. Math. J. 1 31544 74, 4481-4495

work page 2001
[3]

Bandtlow, O. F. and Jenkinson, O. (2008). On the Ruelle eigenvalu e sequence. Ergodic Theory and Dynamical Systems, 28(6), 1701-1711

work page 2008
[4]

”Explicit eigenvalue estim ates for transfer operators acting on spaces of holomorphic functions.” Advances in Mathematics 218.3 (2008): 902-925

Bandtlow, Oscar F., and Oliver Jenkinson. ”Explicit eigenvalue estim ates for transfer operators acting on spaces of holomorphic functions.” Advances in Mathematics 218.3 (2008): 902-925

work page 2008
[5]

Simon, B. (2010). Trace ideals and their applications (No. 120). A merican Mathematical Soc

work page 2010
[6]

Application of the method of moments in proba bility and statistics

Diaconis, P.(1987). Application of the method of moments in proba bility and statistics. In Moments in mathematics (San Antonio, Tex., 1987), volume 37 of Proc. Sympo s. Appl. Math., pages 125-142. Amer. Math. Soc., Providence, RI

work page 1987
[7]

Cipriano, I. (2018). The Wasserstein distance between station ary measures, arXiv:1611.00092v3

work page internal anchor Pith review Pith/arXiv arXiv 2018
[8]

and Pollicott, M

Cipriano, I. and Pollicott, M. (2018). Stationary measures asso ciated to analytic iterated function schemes, Math. Nachr. , 291:7, 1049-1054

work page 2018
[9]

Falconer, K. (1990). Fractal Geometry: Mathematical Found ations and Applications, John Wiley and Sons

work page 1990
[10]

Fan, A. H. and Lau, K.-S. (1999). Iterated Function System a nd Ruelle Operator, J. Math. Anal. & Appl. 231, 319-344

work page 1999
[11]

Fraser, J. (2015). First and second moments for self-couplin gs and Wasserstein distances, Mathemat- ishe Nachrichten , 288, 2028-2041

work page 2015
[12]

Froyland, G. (1999). Ulam’s method for random interval maps, Nonlinearity 12 1029-52 13, 958-985

work page 1999
[13]

Galatolo, S., Monge M., and Nisoli, I. (2016). Rigorous approximat ion of stationary measures and convergence to equilibrium for iterated function systems, Journa l of Physics A: Mathematical and Theoretical, 49:27

work page 2016
[14]

Gohberg, I., Goldberg, S., and Kaashoek, M. A. (2013). Classe s of linear operators I (Vol. 63). Birkh¨ auser

work page 2013
[15]

Gohberg, I., Goldberg, S., and Krupnik, N. (2012). Traces and determinants of linear operators (Vol. 116). Birkh¨ auser

work page 2012
[16]

and Herv´ e, L

Hennion, H. and Herv´ e, L. (2001). Limit Theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness (Vol. 1766). Springer S cience and Business Media

work page 2001
[17]

Hutchinson, J. (1981). Fractals and self similarity, Indiana Univ. Math. J. , 30, 713-747

work page 1981
[18]

Kato, T. (2013). Perturbation theory for linear operators ( Vol. 132). Springer Science and Business Media

work page 2013
[19]

(2019) ”Eﬀective estimates on the to p Lyapunov exponent for random matrix products.” arXiv preprint arXiv:1901.10944

Jurga, N., and Morris, I. (2019) ”Eﬀective estimates on the to p Lyapunov exponent for random matrix products.” arXiv preprint arXiv:1901.10944. 18 ITALO CIPRIANO AND NATALIA JURGA

work page arXiv 2019
[20]

(2005) Algorithms for approximation of invariant measur es for ifs, Manuscripta Mathem- atica 116 31-55

¨Oberg, A. (2005) Algorithms for approximation of invariant measur es for ifs, Manuscripta Mathem- atica 116 31-55

work page 2005
[21]

”A dynamical approach to accelerating numerical integration with equidistributed points.” Proceedings of the Steklov Institute o f Mathematics 256.1 (2007): 275- 289

Jenkinson, Oliver, and Mark Pollicott. ”A dynamical approach to accelerating numerical integration with equidistributed points.” Proceedings of the Steklov Institute o f Mathematics 256.1 (2007): 275- 289

work page 2007
[22]

Jenkinson, Oliver, and Mark Pollicott. ”Rigorous eﬀective bound s on the Hausdorﬀ dimension of continued fraction Cantor sets: a hundred decimal digits for the d imension of E2.” Advances in Math- ematics 325 (2018): 87-115

work page 2018
[23]

Ruelle, D. (1976). Zeta-functions for expanding maps and Ano sov ﬂows. Inventiones mathematicae, 34(3), 231-242

work page 1976
[24]

and Zhang, T

Strichartz, R., Taylor, A. and Zhang, T. (995). Densities of se lf-similar measures on the line, Experi- mental Mathematics, 4, 101-128. F acultad de Matem ´aticas, Pontificia Universidad Cat ´olica de Chile (PUC), A venida Vicu˜na Mackenna 4860, Santiago, Chile E-mail address : icipriano@gmail.com URL: http://www.icipriano.org/ Department of Mathematics,...

work page

[1] [1]

Akhiezer, N. I. (1965). The classical moment problem and some r elated questions in analysis. Translated by N. Kemmer. Hafner Publishing Co., New York

work page 1965

[2] [2]

Blank, M. (2001). Perron-Frobenius spectrum for random map s and its approximation, Mosc. Math. J. 1 31544 74, 4481-4495

work page 2001

[3] [3]

Bandtlow, O. F. and Jenkinson, O. (2008). On the Ruelle eigenvalu e sequence. Ergodic Theory and Dynamical Systems, 28(6), 1701-1711

work page 2008

[4] [4]

”Explicit eigenvalue estim ates for transfer operators acting on spaces of holomorphic functions.” Advances in Mathematics 218.3 (2008): 902-925

Bandtlow, Oscar F., and Oliver Jenkinson. ”Explicit eigenvalue estim ates for transfer operators acting on spaces of holomorphic functions.” Advances in Mathematics 218.3 (2008): 902-925

work page 2008

[5] [5]

Simon, B. (2010). Trace ideals and their applications (No. 120). A merican Mathematical Soc

work page 2010

[6] [6]

Application of the method of moments in proba bility and statistics

Diaconis, P.(1987). Application of the method of moments in proba bility and statistics. In Moments in mathematics (San Antonio, Tex., 1987), volume 37 of Proc. Sympo s. Appl. Math., pages 125-142. Amer. Math. Soc., Providence, RI

work page 1987

[7] [7]

Cipriano, I. (2018). The Wasserstein distance between station ary measures, arXiv:1611.00092v3

work page internal anchor Pith review Pith/arXiv arXiv 2018

[8] [8]

and Pollicott, M

Cipriano, I. and Pollicott, M. (2018). Stationary measures asso ciated to analytic iterated function schemes, Math. Nachr. , 291:7, 1049-1054

work page 2018

[9] [9]

Falconer, K. (1990). Fractal Geometry: Mathematical Found ations and Applications, John Wiley and Sons

work page 1990

[10] [10]

Fan, A. H. and Lau, K.-S. (1999). Iterated Function System a nd Ruelle Operator, J. Math. Anal. & Appl. 231, 319-344

work page 1999

[11] [11]

Fraser, J. (2015). First and second moments for self-couplin gs and Wasserstein distances, Mathemat- ishe Nachrichten , 288, 2028-2041

work page 2015

[12] [12]

Froyland, G. (1999). Ulam’s method for random interval maps, Nonlinearity 12 1029-52 13, 958-985

work page 1999

[13] [13]

Galatolo, S., Monge M., and Nisoli, I. (2016). Rigorous approximat ion of stationary measures and convergence to equilibrium for iterated function systems, Journa l of Physics A: Mathematical and Theoretical, 49:27

work page 2016

[14] [14]

Gohberg, I., Goldberg, S., and Kaashoek, M. A. (2013). Classe s of linear operators I (Vol. 63). Birkh¨ auser

work page 2013

[15] [15]

Gohberg, I., Goldberg, S., and Krupnik, N. (2012). Traces and determinants of linear operators (Vol. 116). Birkh¨ auser

work page 2012

[16] [16]

and Herv´ e, L

Hennion, H. and Herv´ e, L. (2001). Limit Theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness (Vol. 1766). Springer S cience and Business Media

work page 2001

[17] [17]

Hutchinson, J. (1981). Fractals and self similarity, Indiana Univ. Math. J. , 30, 713-747

work page 1981

[18] [18]

Kato, T. (2013). Perturbation theory for linear operators ( Vol. 132). Springer Science and Business Media

work page 2013

[19] [19]

(2019) ”Eﬀective estimates on the to p Lyapunov exponent for random matrix products.” arXiv preprint arXiv:1901.10944

Jurga, N., and Morris, I. (2019) ”Eﬀective estimates on the to p Lyapunov exponent for random matrix products.” arXiv preprint arXiv:1901.10944. 18 ITALO CIPRIANO AND NATALIA JURGA

work page arXiv 2019

[20] [20]

(2005) Algorithms for approximation of invariant measur es for ifs, Manuscripta Mathem- atica 116 31-55

¨Oberg, A. (2005) Algorithms for approximation of invariant measur es for ifs, Manuscripta Mathem- atica 116 31-55

work page 2005

[21] [21]

”A dynamical approach to accelerating numerical integration with equidistributed points.” Proceedings of the Steklov Institute o f Mathematics 256.1 (2007): 275- 289

Jenkinson, Oliver, and Mark Pollicott. ”A dynamical approach to accelerating numerical integration with equidistributed points.” Proceedings of the Steklov Institute o f Mathematics 256.1 (2007): 275- 289

work page 2007

[22] [22]

Jenkinson, Oliver, and Mark Pollicott. ”Rigorous eﬀective bound s on the Hausdorﬀ dimension of continued fraction Cantor sets: a hundred decimal digits for the d imension of E2.” Advances in Math- ematics 325 (2018): 87-115

work page 2018

[23] [23]

Ruelle, D. (1976). Zeta-functions for expanding maps and Ano sov ﬂows. Inventiones mathematicae, 34(3), 231-242

work page 1976

[24] [24]

and Zhang, T

Strichartz, R., Taylor, A. and Zhang, T. (995). Densities of se lf-similar measures on the line, Experi- mental Mathematics, 4, 101-128. F acultad de Matem ´aticas, Pontificia Universidad Cat ´olica de Chile (PUC), A venida Vicu˜na Mackenna 4860, Santiago, Chile E-mail address : icipriano@gmail.com URL: http://www.icipriano.org/ Department of Mathematics,...

work page