arxiv: 2605.01009 · v1 · submitted 2026-05-01 · 🧮 math.DS

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Stability Theory for Local Iterated Function Systems

Elismar R. Oliveira, Paulo Varandas

Pith reviewed 2026-05-09 17:59 UTC · model grok-4.3

classification 🧮 math.DS

keywords local iterated function systemscombinatorial stabilitytopological stabilityconcordant shadowingopen set conditioncode spacegraph-directed IFSbeta-transformations

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The pith

Concordant shadowing implies upper semicontinuity of the local attractor and persistence of the code space for contractive local IFSs, yielding combinatorial stability and topological stability under the open set condition on manifolds of维度

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

The paper develops a stability theory for contractive local iterated function systems on compact metric spaces. It proves that concordant shadowing ensures the upper semicontinuity of the local attractor and the persistence of the code space under perturbations, which serves as a criterion for combinatorial stability. Under the open set condition, combinatorially stable contractive local systems admit a strong form of topological stability, with the converse holding on compact manifolds of dimension at least three. This shows that stability in the local setting is governed by the interplay between contraction and the combinatorial rigidity of the code space, with applications to graph-directed IFSs and pseudogroup actions.

Core claim

The authors prove that concordant shadowing in contractive local IFSs on compact metric spaces implies upper semicontinuity of the local attractor and persistence of the code space, thereby providing a criterion for combinatorial stability under perturbations. Under the open set condition, combinatorially stable contractive local systems are topologically stable, and this implication reverses on compact manifolds of dimension at least three. In particular, contractive graph-directed IFSs are topologically stable, while certain local IFSs derived from beta-transformations are combinatorially unstable.

What carries the argument

Concordant shadowing, a property ensuring that perturbed orbits can be shadowed while respecting the original combinatorial code in the symbolic space of the local IFS.

Load-bearing premise

The local IFSs are contractive on compact metric spaces, with the open set condition required to equate combinatorial and topological stability.

What would settle it

An explicit small perturbation of a combinatorially stable contractive local IFS on a compact three-manifold where the attractor fails to vary upper semicontinuously or the code space does not persist would disprove the main implications.

Figures

Figures reproduced from arXiv: 2605.01009 by Elismar R. Oliveira, Paulo Varandas.

**Figure 3.1.** Figure 3.1: Shadowing scheme Let (yi)i⩾0 be a (a, δ)-pseudo orbit for RY. Then, d(yj+1, gj (yj )) < δ for every j ⩾ 0 and yj ∈ Yaj . Suppose that RY is δ-close to RX. Hence distH(Xaj , Yaj ) < δ and, consequently, for each j ⩾ 0 there exists a point xj ∈ Xaj with d(xj , yj ) < δ. (3.10) By triangular inequality, d(fa0 (x0), x1) ⩽ d(fa0 (x0), ga0 (y0)) + d(ga0 (y0), y1) + d(y1, x1). (3.11) By construction, d(y1, x1) … view at source ↗

**Figure 8.1.** Figure 8.1: Iterations F k X (X) for k = 0, 1, 4, 9 and β = 0.3. Although Theorem B ensures upper semicontinuity of the local attractor, the following geometric example highlights the subtle bifurcation mechanism through which lower semicontinuity may fail. More precisely, we construct an increasing family of restrictions X t whose local attractor At X undergoes transitions near t = 1 4 and t = 3 4 , providing a cl… view at source ↗

**Figure 8.2.** Figure 8.2: Family of local attractors At X . The last one is a sequence of scaled copies of the original Sierpinski gasket. {1, 2, . . . , n} G where G is a finitely generated group (cf. [9]). Furthermore, results by Osipov and Tikhomirov [21], show that the shadowing property for group actions is highly sensitive to the algebraic structure of the acting group, and not merely to the dynamics of individual group ele… view at source ↗

read the original abstract

We develop a stability theory for contractive local IFSs on compact metric spaces. Unlike the classical global setting, local systems may exhibit a richer symbolic and geometric structure, including code spaces that are not of finite type and attractors with endpoints, leading to new mechanisms of instability. We first prove that concordant shadowing implies upper semicontinuity of the local attractor and persistence of the code space, yielding a criterion for combinatorial stability under perturbations. Under the open set condition, we establish a strong form of topological stability for combinatorially stable contractive local systems, and prove the converse implication on compact manifolds of dimension at least three. In particular, we show that contractive graph-directed IFSs are topologically stable. We also construct contractive local IFSs derived from beta-transformations that are combinatorially unstable. These results show that stability in the local setting is governed by the interplay between contraction and the combinatorial rigidity of the code space. Applications to graph-directed IFSs and pseudogroup actions are also given.

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

Extends stability results to local IFS with shadowing criteria and beta-transform instability examples, but full proofs are needed to confirm.

read the letter

This paper builds a stability theory for contractive local iterated function systems on compact metric spaces. The key advance is showing that concordant shadowing implies upper semicontinuity of the local attractor plus persistence of the code space, which gives a combinatorial stability criterion. Under the open set condition they get a strong topological stability result for combinatorially stable systems, plus a converse on manifolds of dimension at least three. They also prove graph-directed IFS are topologically stable and construct combinatorially unstable examples coming from beta-transformations. These pieces highlight how local systems can have non-finite-type code spaces and endpoints that create instability routes not seen in the global case.

Referee Report

0 major / 2 minor

Summary. The paper develops a stability theory for contractive local iterated function systems (IFSs) on compact metric spaces. It proves that concordant shadowing implies upper semicontinuity of the local attractor and persistence of the code space, providing a criterion for combinatorial stability under perturbations. Under the open set condition, it establishes a strong form of topological stability for combinatorially stable contractive local systems and proves the converse on compact manifolds of dimension at least three. In particular, contractive graph-directed IFSs are shown to be topologically stable. The paper also constructs contractive local IFSs derived from beta-transformations that are combinatorially unstable, and discusses applications to graph-directed IFSs and pseudogroup actions. Stability is governed by the interplay between contraction and the combinatorial rigidity of the code space.

Significance. If the results hold, this work meaningfully extends classical global IFS stability theory to the local setting, where code spaces need not be of finite type and attractors may have endpoints, introducing new instability mechanisms. The criteria involving concordant shadowing and the open set condition, together with the explicit result on graph-directed IFSs and the beta-transformation counterexamples, provide concrete, testable advances. The applications to pseudogroup actions further broaden the scope. These contributions are likely to influence research in dynamical systems, fractal geometry, and symbolic dynamics.

minor comments (2)

The abstract and introduction would benefit from a brief, concrete example of a local IFS exhibiting endpoints or a non-finite-type code space early on, to illustrate the richer structure mentioned in the opening paragraph.
Notation for the code space and its persistence under perturbation should be introduced with a short table or diagram in the preliminary section to aid readability for readers unfamiliar with local IFS variants.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on stability theory for contractive local IFSs and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper develops a stability theory for contractive local IFSs via a sequence of theorems: concordant shadowing implies upper semicontinuity of the attractor and persistence of the code space; under the open set condition this yields topological stability for combinatorially stable systems, with a converse on manifolds of dimension at least three. These are standard deductive steps in dynamical systems, grounded in the stated assumptions of contractivity on compact metric spaces and the open set condition. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described claims. The derivations remain independent of the target results and are self-contained within the given hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The theory rests on standard background assumptions from metric space topology and IFS theory; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (3)

standard math The underlying space is a compact metric space
Invoked as the setting for all local IFSs and attractors.
domain assumption The local IFS is contractive
Required for the shadowing and stability statements.
domain assumption The open set condition holds
Used to obtain the strong form of topological stability.

pith-pipeline@v0.9.0 · 5466 in / 1437 out tokens · 50369 ms · 2026-05-09T17:59:29.033083+00:00 · methodology

discussion (0)

Reference graph

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