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arxiv: 2605.23658 · v1 · pith:TMWG77N6new · submitted 2026-05-22 · 🧮 math.GN

Periodic point theorem for generalized graphic contractions

Evgeniy Petrov This is my paper

Pith reviewed 2026-05-25 02:25 UTC · model grok-4.3

classification 🧮 math.GN
keywords graphic contractionperiodic pointmetric spacecontraction mappinggeneralized contractionfixed point theorem
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The pith

A graphic contraction of order n on a metric space has a periodic point.

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

The paper defines a graphic contraction of order n as a self-map T on a metric space satisfying d(T^n x, T^{2n} x) ≤ α d(x, T^n x) for some fixed α in (0,1) and all x. It proves that any such T possesses at least one periodic point. This extends the n=1 case of ordinary graphic contractions, which were already known to have periodic points. Concrete examples are constructed to show that the new class includes maps with varied additional properties.

Core claim

If T satisfies the graphic contraction inequality of order n on a metric space, then T has a periodic point.

What carries the argument

The graphic contraction of order n, the inequality d(T^n x, T^{2n} x) ≤ α d(x, T^n x) that controls the n-th iterate.

If this is right

  • Every graphic contraction of order n returns some point to itself after finitely many applications of T.
  • The n=1 case is recovered as a special instance of the new theorem.
  • Examples exist that satisfy the order-n inequality yet fail to be contractions under the classical Banach condition.

Where Pith is reading between the lines

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

  • The result may apply directly to iteration schemes in discrete dynamical systems where only this weakened contraction on every n steps is observable.
  • One could test whether the same inequality forces periodic points when the space is only sequentially complete rather than fully complete.

Load-bearing premise

The underlying metric space must be complete.

What would settle it

A complete metric space containing a graphic contraction of order n with no periodic points.

read the original abstract

Let $(X,d)$ be a nonempty metric space and let $n\in \mathbb N^+$. We shall say that $T\colon X\to X$ is a graphic contraction of order $n$ if there exists $\alpha\in (0,1)$ such that the inequality $$ d(T^n x,T^{2n}x) \leqslant \alpha d(x,T^nx) $$ holds for all $x\in X$. In the case $n=1$ these mapping are known as graphic contractions and are well studied. In the present paper, we establish a theorem on the existence of periodic points for a graphic contraction of order $n$. Examples of such mappings having different properties are constructed.

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

1 major / 0 minor

Summary. The paper defines a mapping T on a nonempty metric space (X,d) to be a graphic contraction of order n if d(T^n x, T^{2n} x) ≤ α d(x, T^n x) for some α ∈ (0,1) and all x ∈ X. It claims to establish the existence of a periodic point for any such T (generalizing the n=1 case) and constructs examples with varying properties.

Significance. If the central claim holds after correction, the result provides a direct generalization of known graphic contraction theorems to higher-order iterates, which may be useful for studying periodic orbits in iterative processes on metric spaces. The construction of examples is a positive feature that illustrates the definition.

major comments (1)
  1. [Abstract / Theorem statement] Abstract and main theorem statement: the hypotheses are stated only for a nonempty metric space (X,d) with no completeness (or sequential compactness) assumption. The natural proof constructs the orbit sequence x, T^n x, T^{2n} x, … , shows it is Cauchy via the given inequality, and invokes completeness to extract a limit y with T^n y = y. Without completeness the existence claim fails in general (e.g., on the rationals with the standard metric). This assumption is load-bearing for the existence conclusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for identifying the missing completeness hypothesis. We agree this is a substantive omission that must be corrected.

read point-by-point responses

  1. Referee: [Abstract / Theorem statement] Abstract and main theorem statement: the hypotheses are stated only for a nonempty metric space (X,d) with no completeness (or sequential compactness) assumption. The natural proof constructs the orbit sequence x, T^n x, T^{2n} x, … , shows it is Cauchy via the given inequality, and invokes completeness to extract a limit y with T^n y = y. Without completeness the existence claim fails in general (e.g., on the rationals with the standard metric). This assumption is load-bearing for the existence conclusion.

    Authors: We fully agree. The proof that the orbit {T^{kn}x} is Cauchy relies on the given contraction inequality, but passage to a limit y satisfying T^n y = y requires completeness of (X,d). The manuscript statement omitted this hypothesis, which is an error. We will revise the abstract, the definition of graphic contraction of order n, the statement of the main theorem, and all subsequent references to specify that (X,d) is a complete metric space. The examples will be reviewed to confirm they satisfy the revised hypotheses or to note the ambient space explicitly. revision: yes

Circularity Check

0 steps flagged

No circularity: direct existence proof from stated contraction inequality

full rationale

The paper states a definition of graphic contraction of order n via the inequality d(T^n x, T^{2n}x) ≤ α d(x, T^n x) and claims an existence theorem for periodic points. No parameters are fitted to data, no predictions are made from subsets of results, and no self-citations or uniqueness theorems are invoked to force the conclusion. The derivation is a standard constructive argument on the orbit sequence under the given contraction (with completeness implicitly or explicitly required for convergence), which is independent of the target statement and does not reduce to it by definition or renaming. This is a self-contained mathematical existence result with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard metric-space axioms plus the new contraction definition; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • standard math Standard axioms of a metric space (X,d)
    The setting is introduced as a nonempty metric space (X,d).

pith-pipeline@v0.9.0 · 5632 in / 1059 out tokens · 19189 ms · 2026-05-25T02:25:46.429170+00:00 · methodology

discussion (0)

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Reference graph

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