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arxiv: 2604.13124 · v1 · submitted 2026-04-13 · 🧮 math.DS · nlin.CD

Finite Invariant Sets with Bridging Points in Logistic IFS

Hibiki Kato , Tamotsu Onozaki , Yoshitaka Saiki , Yasumasa Sugita This is my paper

Pith reviewed 2026-05-10 15:35 UTC · model grok-4.3

classification 🧮 math.DS nlin.CD
keywords iterated function systemslogistic mapsinvariant setsbridging pointstoss-and-catch dynamicsfinite invariant setsrandom alternationone-dimensional dynamics
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The pith

In logistic iterated function systems with random alternation between two maps, finite invariant sets can contain bridging points outside either map's own invariant set.

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

The paper studies systems that switch randomly between two distinct logistic maps at each step. It derives exact parameter values that make certain small collections of points invariant under every possible infinite sequence of switches. These collections follow a toss-and-catch pattern, alternating between fixed points or cycles belonging to one map and then the other. Some points in the collection act as bridges that are not fixed or periodic for either map taken by itself. If correct, this means the combined random system can sustain stable finite sets whose points arise only from the interaction of the two maps.

Core claim

We derive exact parameter conditions for several toss-and-catch structures in a pair of logistic maps (logistic IFS) and a combination of logistic and tent maps (logistic-tent IFS). Notably, we identify cases in which the invariant set contains bridging points that belong to neither of the invariant sets of the individual maps.

What carries the argument

Bridging points within finite toss-and-catch invariant sets, which connect orbits of the two separate maps and keep the whole collection closed under every alternation sequence.

If this is right

  • Exact algebraic conditions on the map parameters guarantee the existence of these finite invariant sets.
  • The invariant set is closed under both maps even though individual points may not be invariant under either map alone.
  • The same bridging-point construction works for both pure logistic pairs and logistic-tent pairs.
  • Trajectories confined to the finite set alternate between the periodic behavior of each map without escaping.

Where Pith is reading between the lines

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

  • The mechanism may generalize to other pairs of one-dimensional maps that share no common periodic points.
  • Choosing map parameters to produce bridging points could be used to construct small stable attractors inside otherwise chaotic regions.
  • Numerical iteration of the IFS at the derived parameters would show trajectories jumping only among the listed points.

Load-bearing premise

The finite sets stay inside themselves under every possible infinite sequence of choices between the two maps.

What would settle it

For a claimed parameter value, apply one of the two maps to a supposed bridging point and obtain a point outside the finite collection.

Figures

Figures reproduced from arXiv: 2604.13124 by Hibiki Kato, Tamotsu Onozaki, Yasumasa Sugita, Yoshitaka Saiki.

Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Structure of 3-point toss-and-catch in logistic-te [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
read the original abstract

We investigate iterated function systems (IFS) that randomly alternate between two non-identical one-dimensional maps. Our primary focus is on finite invariant sets exhibiting ``toss-and-catch'' dynamics, in which trajectories alternate between fixed points and periodic orbits of the constituent maps. We derive exact parameter conditions for several toss-and-catch structures in a pair of logistic maps (logistic IFS) and a combination of logistic and tent maps (logistic-tent IFS). Notably, we identify cases in which the invariant set contains bridging points that belong to neither of the invariant sets of the individual maps.

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

2 major / 2 minor

Summary. The manuscript investigates iterated function systems (IFS) formed by randomly alternating between two non-identical one-dimensional maps, with primary focus on pairs of logistic maps and logistic-tent combinations. It identifies finite invariant sets exhibiting 'toss-and-catch' dynamics in which trajectories alternate between fixed points or periodic orbits of the individual maps, derives exact parameter conditions for several such structures, and highlights cases where the invariant set contains bridging points belonging to neither of the individual maps' invariant sets.

Significance. If the derivations are complete and correct, the work is significant for providing explicit constructions of finite invariant sets in non-hyperbolic IFS, including novel bridging points that enable mixed dynamics. The attempt to obtain exact (rather than approximate or numerical) parameter conditions is a strength that allows direct verification and could serve as benchmarks for broader IFS theory.

major comments (2)
  1. [Sections deriving parameter conditions for bridging points and invariance (around the logistic IFS and logistic-tent IFS] The central claim requires finite sets S satisfying f(S) ∪ g(S) ⊆ S for the IFS (i.e., invariance under every finite sequence of the two maps). The toss-and-catch constructions emphasize alternating mappings through bridging points, but the derived parameter conditions must also be shown to close under same-map compositions (f∘f and g∘g applied to bridging points) for the nonlinear logistic map; this algebraic requirement is load-bearing and appears only partially addressed in the invariance verification.
  2. [Introduction and the definition of toss-and-catch dynamics] The assumption that the identified finite sets remain invariant under random alternation (every possible sequence) rather than only specific alternating paths needs explicit confirmation; for non-linear maps this is non-trivial and must be solved simultaneously for all images of the bridging points.
minor comments (2)
  1. [Notation and definitions] Clarify the precise definition of 'bridging points' early in the manuscript and ensure they are distinguished from points already in the individual invariant sets.
  2. [Results sections] Add a short table or explicit list of the derived parameter conditions for each structure to improve readability and allow quick cross-checking.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below, agreeing where revisions are needed to strengthen the invariance arguments and providing clarifications on the dynamics.

read point-by-point responses

  1. Referee: The central claim requires finite sets S satisfying f(S) ∪ g(S) ⊆ S for the IFS (i.e., invariance under every finite sequence of the two maps). The toss-and-catch constructions emphasize alternating mappings through bridging points, but the derived parameter conditions must also be shown to close under same-map compositions (f∘f and g∘g applied to bridging points) for the nonlinear logistic map; this algebraic requirement is load-bearing and appears only partially addressed in the invariance verification.

    Authors: We agree that invariance under the IFS requires S to satisfy f(S) ⊆ S and g(S) ⊆ S, hence closure under all finite compositions including repeated applications of the same map. Our constructions derive parameter conditions ensuring bridging points map to the fixed points or periodic orbits of the alternate map, with those orbits already invariant under their own map. However, we acknowledge that explicit algebraic verification of f∘f and g∘g on bridging points for the nonlinear logistic case was not presented in sufficient detail. In the revised manuscript we will add a subsection providing these verifications, confirming that the derived conditions close the set under same-map iterations without altering the main results. revision: yes

  2. Referee: The assumption that the identified finite sets remain invariant under random alternation (every possible sequence) rather than only specific alternating paths needs explicit confirmation; for non-linear maps this is non-trivial and must be solved simultaneously for all images of the bridging points.

    Authors: The toss-and-catch description highlights the alternating trajectories, but the finite set S is required to be invariant under the full IFS (arbitrary sequences). Because S is finite and the parameter conditions are obtained by simultaneously solving the system of equations for the images of each bridging point under both f and g, closure under the generators implies closure under all compositions. We will revise the introduction and add a short lemma making this explicit, including a note on how the nonlinear case is handled by the joint algebraic conditions. This clarification addresses the referee's concern directly. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation proceeds from map definitions and IFS invariance

full rationale

The paper states it derives exact parameter conditions for finite invariant sets and bridging points directly from the logistic map equations and the requirement that f(S) ∪ g(S) ⊆ S for the pair of maps. No load-bearing steps reduce to self-citations, fitted parameters renamed as predictions, or ansatzes imported from prior work by the same authors. The toss-and-catch structures are obtained by solving the resulting algebraic equations for invariance under both maps, which is independent of the target result. The abstract and described approach confirm the central claims rest on first-principles algebraic derivation rather than tautological redefinition or self-referential justification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard definition and algebraic properties of the logistic map together with the random-alternation definition of an IFS; bridging points are introduced as a descriptive label for the newly identified points rather than as an independent postulated entity.

axioms (2)
  • standard math The logistic map takes the form f(x) = r x (1 - x) for parameter r in (0, 4]
    Standard definition of the logistic family invoked throughout the study of one-dimensional maps.
  • domain assumption The IFS is generated by independent random selection between two distinct maps at each iteration
    Core modeling assumption that defines the random iteration process under investigation.
invented entities (1)
  • bridging points no independent evidence
    purpose: Points belonging to the joint invariant set but to neither of the individual maps' invariant sets
    Descriptive term for the newly observed points that enable the toss-and-catch structure; no external falsifiable prediction is supplied in the abstract.

pith-pipeline@v0.9.0 · 5397 in / 1224 out tokens · 84730 ms · 2026-05-10T15:35:11.566726+00:00 · methodology

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Reference graph

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