Non-stationary Fractal Interpolation

Peter Massopust

arxiv: 1907.00627 · v1 · pith:YIG6EFEPnew · submitted 2019-07-01 · 🧮 math.DS · math.MG

Non-stationary Fractal Interpolation

Peter Massopust This is my paper

Pith reviewed 2026-05-25 11:37 UTC · model grok-4.3

classification 🧮 math.DS math.MG

keywords non-stationary iterated function systemsfractal interpolationfractal functionsset-valued mapsfixed pointstrajectoriesiterated function systems

0 comments

The pith

Non-stationary iterated function systems generate fractal functions whose local and global behaviors can differ.

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

The paper defines non-stationary iterated function systems through a countable sequence of distinct set-valued maps, each generated by its own iterated function system. It invokes non-stationary fixed-point results and forward-backward trajectories to construct attractors that serve as fractal functions. These functions are shown to support interpolation while permitting properties to change from one region or scale to another. The approach therefore widens the range of data sets or geometric objects that fractal methods can represent. A reader would see the value in having interpolation tools that no longer require the same map to be repeated at every step.

Core claim

By replacing a single fixed iterated function system with a countable sequence of distinct set-valued maps {F_k}, and applying the theory of non-stationary fixed points together with forward and backward trajectories, one obtains new classes of fractal functions that exhibit different local and global behavior and thereby extend fractal interpolation to a non-stationary setting.

What carries the argument

Non-stationary iterated function system: a sequence {F_k} of distinct set-valued maps on the hyperspace H(X), each arising from an iterated function system, whose attractors are obtained via forward and backward trajectories.

If this is right

Fractal functions can now be built whose scaling properties change from one interval to the next.
Interpolation problems can be solved when the underlying contraction maps themselves vary with the iteration index.
Attractors are determined by the entire sequence of maps rather than by a single repeated map.
Local and global regularity can be controlled separately by choosing different maps in different parts of the sequence.

Where Pith is reading between the lines

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

The construction may extend to modeling signals whose statistics evolve over time by letting the choice of map depend on an external parameter.
Numerical schemes could adapt the sequence of maps on the fly to match observed data variation without recomputing a global fixed point.
Links may exist to non-autonomous dynamical systems in which the driving map changes at each discrete time step.

Load-bearing premise

The theory of non-stationary fixed points applies directly to a countable sequence of distinct set-valued maps coming from iterated function systems.

What would settle it

An explicit sequence of distinct IFS maps whose generated attractor either fails to exist, fails to interpolate given data points, or shows identical rather than differing local and global behavior.

Figures

Figures reproduced from arXiv: 1907.00627 by Peter Massopust.

**Figure 1.** Figure 1: The hybrid τ − q attractor. It is smooth at one scale but fractal at another. 6. non-stationary Fractal Interpolation Let us now consider the case X := [0, 1] and Y := R. Both spaces are metrizable under the usual Euclidean distance. In the following, we consider a sequence {Tk} of RB operators of the form (4.18) acting on an appropriate metric subspace of B[0, 1] := B([0, 1], R). Our emphasis here lies in… view at source ↗

**Figure 2.** Figure 2: The hybrid Kiesswetter-Casino attractor. Remark 6.2. Theorem 4.1 holds in the case of non-stationary fractal functions as well. For k ∈ N, a non-stationary IFS is associated with Tk by setting wik,k(x, y) := (lik,k(x), f ◦ lik,k(x) + Sik,k(x) · (y − b)). The conditions imposed on Sik,k and the form of the second component allows the immediate transfer of the proof of Theorem 4.1. Hence, even in the nonst… view at source ↗

read the original abstract

We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps $\{\mathcal{F}_k\}_{k\in \mathbb{N}}$ where each $\mathcal{F}_k$ maps $\mathcal{H}(X)\to \mathcal{H}(X)$ and arises from an iterated function system. Employing the recently developed theory of non-stationary versions of fixed points [11] and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior, and extend fractal interpolation to this new, more flexible setting.

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 defines non-stationary IFS via a sequence of distinct maps and invokes [11] for new fractal interpolants, but leaves the required verification of the fixed-point hypotheses unaddressed.

read the letter

The main point is that Massopust replaces the usual fixed IFS with a countable sequence of different set-valued maps F_k and then applies non-stationary fixed-point results from [11] to obtain attractors for fractal interpolation that can vary locally and globally. The setup itself is stated cleanly: each F_k comes from an IFS on the hyperspace, and forward/backward trajectories are used to handle the changing maps. That is a direct and reasonable extension of the stationary case, and it fits the existing literature on fractal functions without obvious circularity. The citation to [11] is appropriate for the non-stationary part, and the overall construction reads as a natural next step rather than a relabeling. The soft spot sits exactly where the stress-test note flags it. The abstract assumes the hypotheses of [11] (contraction in the Hausdorff metric, trajectory conditions, uniformity across the sequence) carry over to these particular IFS-derived maps, but supplies no check or adaptation. Without that step, or at least an example showing the new behavior, the claim that the construction produces genuinely new classes of functions rests on an unconfirmed transfer. No concrete examples appear in the given text, which makes it harder to assess the practical difference from stationary interpolants. This is for specialists already working in fractal interpolation and non-stationary dynamical systems on metric spaces. A reader who knows both the classical Barnsley theory and the paper cited as [11] would be the right audience and could extract value from seeing the two combined. It deserves a serious referee because the idea is coherent on its own terms and the subfield is small enough that a careful verification of the fixed-point application would be useful. I would send it to review, expecting the authors to supply the missing check that the conditions from [11] hold or to add the necessary adjustments.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces non-stationary iterated function systems consisting of a countable sequence of distinct set-valued maps {F_k} each arising from an iterated function system. It invokes the non-stationary fixed-point theory of reference [11] together with forward and backward trajectories to construct new classes of fractal functions that exhibit different local and global behavior and to extend fractal interpolation to this setting.

Significance. If the transfer of the non-stationary fixed-point results to the IFS-derived maps {F_k} is valid, the construction would supply a more flexible framework for generating fractal functions whose local and global properties can vary across iterations, extending the classical stationary theory in a natural way. The explicit reliance on an external theorem is noted as a potential strength provided the hypotheses are verified.

major comments (2)

[Abstract / main construction] Abstract and the statement of the main construction: the manuscript asserts that the non-stationary fixed-point results of [11] apply directly to the countable sequence of distinct set-valued maps {F_k}:H(X)→H(X) obtained from iterated function systems, yet supplies no verification that the requisite contraction conditions in the Hausdorff metric, forward/backward trajectory requirements, or any uniformity conditions across the varying F_k hold in this setting. This verification is load-bearing for the claimed extension to new fractal functions and interpolation.
[Abstract / introduction] The central claim that the new classes exhibit 'different local and global behavior' rests on the unverified applicability of [11]; without an explicit check or counter-example showing that the IFS maps satisfy the hypotheses of the cited theorem, the extension cannot be regarded as established.

minor comments (2)

[Abstract] The abstract refers to 'new classes of fractal functions' without indicating, even at a high level, how the local versus global behavior differs from the classical stationary case or from each other.
[Abstract] Notation for the sequence {F_k} is introduced but the precise manner in which each F_k is constructed from an underlying IFS (e.g., the choice of contractions or the dependence on k) is not clarified in the provided abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the identification of points requiring clarification. The comments correctly highlight that the manuscript invokes the non-stationary fixed-point theory of [11] without an explicit verification that the IFS-derived maps {F_k} satisfy the necessary hypotheses. We address each major comment below and will revise the manuscript to supply the missing verification.

read point-by-point responses

Referee: [Abstract / main construction] Abstract and the statement of the main construction: the manuscript asserts that the non-stationary fixed-point results of [11] apply directly to the countable sequence of distinct set-valued maps {F_k}:H(X)→H(X) obtained from iterated function systems, yet supplies no verification that the requisite contraction conditions in the Hausdorff metric, forward/backward trajectory requirements, or any uniformity conditions across the varying F_k hold in this setting. This verification is load-bearing for the claimed extension to new fractal functions and interpolation.

Authors: We agree that the manuscript does not contain an explicit check that the sequence {F_k} meets the hypotheses of the cited theorem from [11]. Each F_k is the Hutchinson operator associated to an IFS consisting of contractions, and is therefore a contraction on (H(X), d_H) with ratio equal to the maximum contraction ratio of its constituent maps. In the revision we will insert a new subsection that (i) recalls the precise statement of the relevant result from [11], (ii) verifies that each individual F_k is a contraction, and (iii) states the additional conditions on the sequence of contraction ratios that guarantee the existence of the required forward and backward trajectories. These conditions will be formulated as explicit assumptions on the IFS data. revision: yes
Referee: [Abstract / introduction] The central claim that the new classes exhibit 'different local and global behavior' rests on the unverified applicability of [11]; without an explicit check or counter-example showing that the IFS maps satisfy the hypotheses of the cited theorem, the extension cannot be regarded as established.

Authors: The claim of differing local and global behavior is intended to follow once the non-stationary fixed-point theorem applies. Because the verification step is absent, the referee’s observation is accurate. The revised manuscript will therefore include both the verification described above and a brief discussion (with a simple example) illustrating how the varying contraction ratios across the sequence {F_k} produce functions whose local Hölder exponents or global regularity differ from those obtained by any single stationary IFS. If the referee prefers, we can also supply a counter-example showing what fails when the uniformity conditions on the ratios are violated. revision: yes

Circularity Check

0 steps flagged

No circularity; relies on external cited theory [11] applied to IFS maps

full rationale

The paper defines non-stationary IFS via a countable sequence of distinct set-valued maps {F_k} arising from iterated function systems, then invokes the non-stationary fixed-point theory from reference [11] (described as recently developed and external) together with forward/backward trajectories to construct new fractal functions and extend interpolation. No self-definitional steps appear, no parameters are fitted to data and then renamed as predictions, and no uniqueness theorems or ansatzes are imported via self-citation chains. The central claim is an application of the cited external result rather than a reduction of the output to the paper's own inputs by construction. This is the normal case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of non-stationary fixed-point theory [11] to sequences of distinct IFS maps; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption The theory of non-stationary versions of fixed points developed in [11] applies to countable sequences of distinct set-valued maps arising from iterated function systems.
Invoked in the abstract to guarantee existence of the attractors for the non-stationary systems.

pith-pipeline@v0.9.0 · 5608 in / 1124 out tokens · 27982 ms · 2026-05-25T11:37:55.379699+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 introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps {F_k} ... backward trajectories {Ψ_k(A_0)} converge ... to a unique attractor A
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Lip(F_k) = max{s_{i,k}} < 1 and ∑_{k=1}^∞ ∏_{j=1}^k Lip(F_j) < ∞

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

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

Sobolev Spaces, 2nd ed., Academic Press: New York, 2003

Adams, R., Fourier, J. Sobolev Spaces, 2nd ed., Academic Press: New York, 2003

work page 2003
[2]

Fractals Everywhere, Academic Press: Orlando, USA, 1988

Barnsley, M.F. Fractals Everywhere, Academic Press: Orlando, USA, 1988

work page 1988
[3]

Fractal functions and interpolation

Barnsley, M.F. Fractal functions and interpolation. Constr. Approx. 1986, 2, 303– 329

work page 1986
[4]

Numerics and Fractals

Barnsley, M.F., Hegland, M., Massopust, P.R. Numerics and Fractals. Bull. Inst. Math. Acad. Sin. (N.S.) 2014, 9(3), 389–430. 16 PETER R. MASSOPUST

work page 2014
[5]

V -variable fractals: Fractals with partial self-similarity

Barnsley, M.F.; Hutchinson, J.E.; Stenﬂo, ¨O. V -variable fractals: Fractals with partial self-similarity. Adv. Math. 2008, 218(6), 2015–2088

work page 2008
[6]

Local average sampling and reconstruction with fundamental splines of fractional order

Dira, N., Levin, D., Massopust, P. Attractors of trees of maps and of se- quences of maps between spaces and applications to subdivision. arxiv.org 2019, http://arxiv.org/abs/1904.03434, 1–21

work page internal anchor Pith review Pith/arXiv arXiv 2019
[7]

Inequalities for Stochastic Processes , Dover Publications: New York, 1976

Dubins, L.E., Savage, L.J. Inequalities for Stochastic Processes , Dover Publications: New York, 1976

work page 1976
[8]

Horv´ ath, J.Topological Vector Spaces and Distributions, Addison-Wesley Publishing Company: Reading, USA, 1966

work page 1966
[9]

Fractals and self-similarity

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

work page 1981
[10]

Ein einfaches Beispiel f¨ ur eine Funktion welche ¨ uberall stetig und nicht diﬀerenzierbar ist

Kiesswetter, K. Ein einfaches Beispiel f¨ ur eine Funktion welche ¨ uberall stetig und nicht diﬀerenzierbar ist. Math. Phys. Semesterber. 1966, 13, 216–221

work page 1966
[11]

Non-stationary versions of ﬁxed-point theory, with applications to fractals and subdivision

Levin, D.; Dyn, N.; Viswanathan, P. Non-stationary versions of ﬁxed-point theory, with applications to fractals and subdivision. J. Fixed Point Theory Appl. 2019, 21, 1–25

work page 2019
[12]

Fractal functions and their applications

Massopust, P.R. Fractal functions and their applications. Chaos, Solitons and Frac- tals, 1997, 8(2), 171–190

work page 1997
[13]

Interpolation and Approximation with Splines and Fractals, Oxford University Press: Oxford, USA, 2010

Massopust, P.R. Interpolation and Approximation with Splines and Fractals, Oxford University Press: Oxford, USA, 2010

work page 2010
[14]

Fractal Functions, Fractal Surfaces, and Wavelets , 2nd ed., Aca- demic Press: San Diego, USA, 2016

Massopust, P.R. Fractal Functions, Fractal Surfaces, and Wavelets , 2nd ed., Aca- demic Press: San Diego, USA, 2016

work page 2016
[15]

Metric Linear Spaces, Kluwer Academic: Warsaw, Poland, 1985

Rolewicz, S. Metric Linear Spaces, Kluwer Academic: Warsaw, Poland, 1985

work page 1985
[16]

Functional Analysis, McGraw–Hill: New York, 1991

Rudin, W. Functional Analysis, McGraw–Hill: New York, 1991

work page 1991
[17]

A simple example of the continuous function without derivative

Takagi, T. A simple example of the continuous function without derivative. Proc. Phys. Math. Soc. Japan 1903, 1, 176–177. Centre of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching b. Munich, Germany E-mail address: massopust@ma.tum.de

work page 1903

[1] [1]

Sobolev Spaces, 2nd ed., Academic Press: New York, 2003

Adams, R., Fourier, J. Sobolev Spaces, 2nd ed., Academic Press: New York, 2003

work page 2003

[2] [2]

Fractals Everywhere, Academic Press: Orlando, USA, 1988

Barnsley, M.F. Fractals Everywhere, Academic Press: Orlando, USA, 1988

work page 1988

[3] [3]

Fractal functions and interpolation

Barnsley, M.F. Fractal functions and interpolation. Constr. Approx. 1986, 2, 303– 329

work page 1986

[4] [4]

Numerics and Fractals

Barnsley, M.F., Hegland, M., Massopust, P.R. Numerics and Fractals. Bull. Inst. Math. Acad. Sin. (N.S.) 2014, 9(3), 389–430. 16 PETER R. MASSOPUST

work page 2014

[5] [5]

V -variable fractals: Fractals with partial self-similarity

Barnsley, M.F.; Hutchinson, J.E.; Stenﬂo, ¨O. V -variable fractals: Fractals with partial self-similarity. Adv. Math. 2008, 218(6), 2015–2088

work page 2008

[6] [6]

Local average sampling and reconstruction with fundamental splines of fractional order

Dira, N., Levin, D., Massopust, P. Attractors of trees of maps and of se- quences of maps between spaces and applications to subdivision. arxiv.org 2019, http://arxiv.org/abs/1904.03434, 1–21

work page internal anchor Pith review Pith/arXiv arXiv 2019

[7] [7]

Inequalities for Stochastic Processes , Dover Publications: New York, 1976

Dubins, L.E., Savage, L.J. Inequalities for Stochastic Processes , Dover Publications: New York, 1976

work page 1976

[8] [8]

Horv´ ath, J.Topological Vector Spaces and Distributions, Addison-Wesley Publishing Company: Reading, USA, 1966

work page 1966

[9] [9]

Fractals and self-similarity

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

work page 1981

[10] [10]

Ein einfaches Beispiel f¨ ur eine Funktion welche ¨ uberall stetig und nicht diﬀerenzierbar ist

Kiesswetter, K. Ein einfaches Beispiel f¨ ur eine Funktion welche ¨ uberall stetig und nicht diﬀerenzierbar ist. Math. Phys. Semesterber. 1966, 13, 216–221

work page 1966

[11] [11]

Non-stationary versions of ﬁxed-point theory, with applications to fractals and subdivision

Levin, D.; Dyn, N.; Viswanathan, P. Non-stationary versions of ﬁxed-point theory, with applications to fractals and subdivision. J. Fixed Point Theory Appl. 2019, 21, 1–25

work page 2019

[12] [12]

Fractal functions and their applications

Massopust, P.R. Fractal functions and their applications. Chaos, Solitons and Frac- tals, 1997, 8(2), 171–190

work page 1997

[13] [13]

Interpolation and Approximation with Splines and Fractals, Oxford University Press: Oxford, USA, 2010

Massopust, P.R. Interpolation and Approximation with Splines and Fractals, Oxford University Press: Oxford, USA, 2010

work page 2010

[14] [14]

Fractal Functions, Fractal Surfaces, and Wavelets , 2nd ed., Aca- demic Press: San Diego, USA, 2016

Massopust, P.R. Fractal Functions, Fractal Surfaces, and Wavelets , 2nd ed., Aca- demic Press: San Diego, USA, 2016

work page 2016

[15] [15]

Metric Linear Spaces, Kluwer Academic: Warsaw, Poland, 1985

Rolewicz, S. Metric Linear Spaces, Kluwer Academic: Warsaw, Poland, 1985

work page 1985

[16] [16]

Functional Analysis, McGraw–Hill: New York, 1991

Rudin, W. Functional Analysis, McGraw–Hill: New York, 1991

work page 1991

[17] [17]

A simple example of the continuous function without derivative

Takagi, T. A simple example of the continuous function without derivative. Proc. Phys. Math. Soc. Japan 1903, 1, 176–177. Centre of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching b. Munich, Germany E-mail address: massopust@ma.tum.de

work page 1903