On the Nonexistence of Continuous Immersions for Discrete-time Systems

Eduardo Sontag; Eron Ristich; Necmiye Ozay

arxiv: 2605.15161 · v2 · pith:ZTTSBTT6new · submitted 2026-05-14 · 📡 eess.SY · cs.SY· math.DS

On the Nonexistence of Continuous Immersions for Discrete-time Systems

Eron Ristich , Eduardo Sontag , Necmiye Ozay This is my paper

Pith reviewed 2026-05-15 02:58 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.DS

keywords discrete-time dynamical systemscontinuous immersionsomega-limit setslinear system theorynonexistencealpha-limit setsnonlinear dynamics

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The pith

Discrete-time systems with countably many but more than one omega-limit sets cannot be continuously and injectively immersed into finite-dimensional linear systems.

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

The paper proves that discrete-time nonlinear dynamical systems whose omega-limit sets are countably infinite in number but not a singleton cannot be immersed into a finite-dimensional linear system by any continuous one-to-one mapping. This extends an earlier obstruction result from continuous-time dynamics to the discrete setting, where the same topological incompatibility between multiple limit sets and linear structure persists. Because such an immersion would otherwise let linear-system tools be applied directly to the nonlinear dynamics, the nonexistence result identifies a concrete barrier for analysis and control. The authors also treat the analogous case of alpha-limit sets and supply examples that make the obstruction explicit.

Core claim

If a discrete-time system has more than one but at most countably many omega-limit sets, then no continuous injective immersion into the state space of a finite-dimensional linear system exists. The argument adapts topological properties of limit sets under continuous injections to the discrete-time iteration, showing that linearity forces a contradiction once multiple distinct limit sets are present. The same nonexistence holds when the sets in question are alpha-limit sets.

What carries the argument

Continuous one-to-one immersion mapping that must preserve discrete-time iteration while sending the nonlinear state space into a linear one.

If this is right

The same obstruction applies when alpha-limit sets replace omega-limit sets.
Any discrete-time system with multiple distinct omega-limit sets lies outside the class that can be analyzed via linear immersion.
Linear-system methods cannot be applied through immersion to systems whose attractors form a countably infinite but non-singleton collection.
The result supplies concrete examples where the nonexistence is immediate.

Where Pith is reading between the lines

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

Relaxing continuity might allow immersions for some of these systems, though the paper does not investigate that relaxation.
The obstruction could extend to switched or hybrid discrete systems whose modes produce multiple limit sets.
Immersions into infinite-dimensional linear systems remain conceivable but are outside the paper's finite-dimensional scope.

Load-bearing premise

Any candidate immersion is required to be both continuous and one-to-one, while the system possesses countably many distinct omega-limit sets.

What would settle it

Explicit construction of a continuous injective mapping that immerses a concrete discrete-time system known to have multiple omega-limit sets into the trajectories of some finite-dimensional linear system.

Figures

Figures reproduced from arXiv: 2605.15161 by Eduardo Sontag, Eron Ristich, Necmiye Ozay.

read the original abstract

Understanding when linear immersions of nonlinear dynamical systems exist is important since such immersions allow us to leverage the rich tools of linear system theory to analyze nonlinear dynamics. Recently, Liu et al. (2023) showed that continuous-time dynamical systems that admit countably many but more than one omega-limit sets cannot be immersed into finite dimensional linear systems with a one-to-one and continuous mapping. In this paper, we extend these results to discrete-time dynamics and show that similar obstructions exist also in discrete time. We further consider a generalization involving alpha-limit sets. Several examples are provided to demonstrate the results.

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

Paper shows nonexistence of continuous immersions for discrete-time systems with countably many omega-limit sets, extending Liu et al. cleanly.

read the letter

The main takeaway is that discrete-time nonlinear systems with countably many but more than one omega-limit set cannot admit a continuous injective immersion into a finite-dimensional linear system. The authors extend the 2023 Liu et al. continuous-time obstruction to the discrete setting and add a version for alpha-limits, with examples to show the claim in action. The argument adapts the topological fact that such an immersion would map distinct omega-limits to distinct ones under the linear map A, yet finite-dimensional linear maps produce either a single omega-limit or uncountably many, never exactly countably many isolated ones. This carries over to iterations without obvious breakage, and the examples illustrate systems that hit the forbidden count. The work is useful because it gives a concrete negative criterion that stops people from trying impossible linear embeddings in sampled-data or discrete control problems. The reasoning stays grounded in standard limit-set properties and avoids circular definitions or fitted parameters. Soft spots are minor. The abstract is brief on the exact immersion conditions and proof steps, so one would check in the full text how injectivity and continuity interact with discrete orbits to separate the limits. The examples need to be verified as truly having countably many omega-limits without slipping into uncountable cases, but nothing in the described logic contradicts itself. The citation pattern is direct and appropriate. This is for control theorists and dynamical-systems people working on immersion methods for discrete or sampled nonlinear systems. It supplies a practical obstruction result rather than a new construction. I would send it to peer review; the extension is focused, the topological core is reproducible, and the negative result is worth referee scrutiny even if revisions tighten the examples or definitions.

Referee Report

2 major / 2 minor

Summary. The paper extends the nonexistence result of Liu et al. (2023) from continuous-time to discrete-time nonlinear systems: a continuous injective immersion h into a finite-dimensional linear system cannot exist when the discrete dynamics possess countably many but more than one distinct ω-limit sets. The argument relies on the fact that such an h would map ω-limit sets injectively to ω-limit sets of the linear map, which cannot realize exactly countably many distinct attractors. A parallel obstruction is derived for α-limit sets, and several concrete discrete-time examples are supplied to illustrate the claim.

Significance. If the topological argument holds, the result supplies a clean, parameter-free obstruction that limits the applicability of linear immersion techniques to discrete-time nonlinear systems possessing multiple countable limit sets. It directly parallels the continuous-time case, supplies explicit examples, and thereby sharpens the boundary between systems that can and cannot be analyzed via finite-dimensional linear models.

major comments (2)

[§3] §3, proof of Theorem 1: the step asserting that a continuous injective h maps distinct ω-limit sets of the nonlinear system to distinct ω-limit sets of the linear map A is load-bearing; the manuscript should explicitly invoke the invariance of ω-limit sets under the discrete iteration and confirm that the target space is Hausdorff so that distinct closed sets remain separated after the image.
[§4] §4, Theorem 2 (α-limit generalization): the argument assumes the immersion is globally defined and one-to-one on the whole state space; if the result is intended only for global immersions, this should be stated explicitly, as local immersions might evade the countable-limit-set obstruction.

minor comments (2)

[Introduction] The abstract and introduction cite Liu et al. (2023) but do not give the precise statement of the continuous-time theorem being extended; adding a one-sentence recap would improve readability.
[Examples] In the examples, the state space and the precise form of the discrete map f should be written with explicit coordinates rather than relying on verbal description alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript extending the nonexistence result to discrete-time systems. We have revised the paper to incorporate the suggested clarifications in the proofs and theorem statements. Our point-by-point responses follow.

read point-by-point responses

Referee: [§3] §3, proof of Theorem 1: the step asserting that a continuous injective h maps distinct ω-limit sets of the nonlinear system to distinct ω-limit sets of the linear map A is load-bearing; the manuscript should explicitly invoke the invariance of ω-limit sets under the discrete iteration and confirm that the target space is Hausdorff so that distinct closed sets remain separated after the image.

Authors: We agree that this step requires additional explicit justification. In the revised manuscript, we have added a sentence invoking the forward-invariance of ω-limit sets under the discrete iteration map and noted that the codomain (finite-dimensional Euclidean space) is Hausdorff. This ensures that the continuous injective image of distinct compact ω-limit sets remains a pair of distinct compact sets, preserving the contradiction with the linear dynamics. The change is confined to a short clarifying paragraph in the proof of Theorem 1. revision: yes
Referee: [§4] §4, Theorem 2 (α-limit generalization): the argument assumes the immersion is globally defined and one-to-one on the whole state space; if the result is intended only for global immersions, this should be stated explicitly, as local immersions might evade the countable-limit-set obstruction.

Authors: We concur. Theorem 2 and its proof are formulated for globally defined continuous injective immersions. In the revised version we have inserted an explicit sentence in the statement of Theorem 2 and in the opening paragraph of §4 clarifying that the result applies only to global immersions and that the obstruction may not hold for merely local immersions. No other changes to the argument were required. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation adapts the topological obstruction from Liu et al. (2023) to discrete time by showing that a continuous injective immersion h would map distinct omega-limit sets of the nonlinear system to distinct omega-limit sets of the target linear system A. Finite-dimensional linear maps realize either a single omega-limit set or uncountably many (e.g., concentric circles under rotation), so countably many but more than one is impossible. This uses only standard facts about omega-limit sets and continuity of h, with no fitted parameters, self-definitional equations, or load-bearing self-citations. The alpha-limit generalization and examples follow identically. The central claim therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard topological and dynamical-systems properties of omega- and alpha-limit sets together with the definition of continuous immersion; no new free parameters or invented entities are introduced.

axioms (2)

standard math Omega-limit sets are nonempty, compact, and invariant for continuous maps on metric spaces.
Invoked when characterizing the obstruction to injectivity of the immersion.
domain assumption A continuous one-to-one immersion into a linear system preserves the number and separation of limit sets.
Central to the nonexistence argument.

pith-pipeline@v0.9.0 · 5401 in / 1158 out tokens · 38550 ms · 2026-05-15T02:58:48.194508+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

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

Suppose that conditions (T2′) and (T3′) hold. If X contains more than one ω-limit set, then this system on X cannot be immersed by a continuous one-to-one immersion F in a system with closed basins
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

F(X) is a disjoint union of a countable collection of closed sets... only one of the sets... can be nonempty

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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.