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arxiv: 2604.13189 · v1 · submitted 2026-04-14 · 🧮 math.DS

The interplay between partial specification, average shadowing, and Besicovitch completeness

Melih Emin Can , Marcin Kulczycki This is my paper

Pith reviewed 2026-05-10 13:52 UTC · model grok-4.3

classification 🧮 math.DS
keywords partial specification propertyaverage shadowing propertyBesicovitch completenessergodic measuresinvariant measurestopological dynamicscompact dynamical systemssurjectivity
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The pith

Compact dynamical systems with the partial specification property also have the average shadowing property.

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

This paper proves that the partial specification property in a compact dynamical system implies the average shadowing property. When the system is additionally surjective, the ergodic measures are dense in the set of all invariant measures. The work also provides an example of a compact dynamical system that fails to be Besicovitch complete. These connections help link different notions of orbit approximation and measure approximation in dynamics.

Core claim

The author proves that partial specification entails average shadowing for any compact dynamical system (X, T). Moreover, if T is surjective, then the ergodic measures are dense in the space of T-invariant probability measures. The paper includes a concrete example of a compact system without Besicovitch completeness to illustrate the boundaries of these properties.

What carries the argument

The partial specification property, which allows finite orbit segments to be specified with arbitrary gaps, is the central mechanism that forces average shadowing and density of ergodics.

Load-bearing premise

The dynamical system is assumed to be a compact metric space with a continuous self-map and the properties are defined via standard metric notions of orbit segments and pseudo-orbits.

What would settle it

Constructing or verifying a compact dynamical system that has partial specification yet lacks average shadowing would directly falsify the main implication.

read the original abstract

Let $(X,T)$ be a compact dynamical system. This article proves that if $(X,T)$ has the partial specification property, then it has the average shadowing property. It is also proven that if $(X,T)$ is surjective and has the partial specification property, then the set of ergodic measures of $(X,T)$ is dense in the space of its invariant measures. An example of a compact dynamical system that is not Besicovitch complete is 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.

Referee Report

0 major / 3 minor

Summary. The paper claims that for a compact dynamical system (X,T), the partial specification property implies the average shadowing property. It further claims that if (X,T) is surjective, then the ergodic measures are dense in the space of invariant measures. An explicit example of a compact dynamical system that is not Besicovitch complete is constructed.

Significance. If correct, the results connect partial specification to average shadowing via explicit orbit-concatenation arguments and give a surjectivity-based criterion for ergodic density, both of which are useful in topological dynamics. The paper uses standard metric-space definitions without hidden uniformity or expansivity assumptions and supplies an explicit counterexample; these are strengths.

minor comments (3)
  1. [§2] §2 (definitions): the precise quantifiers on the gaps in the partial specification property should be restated explicitly before the proof of the average-shadowing implication, to make the block-concatenation construction easier to follow.
  2. [Theorem on ergodic density] Theorem on ergodic density: the role of surjectivity is invoked only for the second claim; a brief remark on whether the density conclusion can fail without surjectivity would clarify the necessity of the hypothesis.
  3. [Example] Example of non-Besicovitch system: include a short verification that the constructed space is compact and that the failure of Besicovitch completeness is witnessed by a concrete sequence of points.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its contributions connecting partial specification to average shadowing and providing a surjectivity criterion for ergodic density, and the recommendation for minor revision. We will incorporate improvements to clarity and presentation in the revised version.

Circularity Check

0 steps flagged

No significant circularity; proofs are direct constructions from definitions

full rationale

The paper establishes two main implications by explicit constructions: partial specification (finite orbit segments with controlled gaps) is used to build average-shadowing pseudo-orbits whose average distance vanishes, and surjectivity plus partial specification yields dense ergodic measures via concatenation of specification blocks. These steps invoke only the stated metric-space definitions and standard compactness; no equation reduces a conclusion to a fitted input, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled. The non-Besicovitch example is constructed independently and does not affect the positive results. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works entirely within the standard axiomatic framework of compact metric dynamical systems; no new entities or fitted constants are introduced.

axioms (1)
  • standard math Compact metric space with continuous map T
    The setup (X,T) compact dynamical system is the usual background assumed in the field.

pith-pipeline@v0.9.0 · 5366 in / 1144 out tokens · 36264 ms · 2026-05-10T13:52:20.905695+00:00 · methodology

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Reference graph

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