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arxiv: 2606.28283 · v1 · pith:C43X4JP2new · submitted 2026-06-26 · 🧮 math.DS

A flexibility result for polynomial entropy of pointwise periodic homeomorphisms

Ma\v{s}a {\DJ}ori\'c , Jelena Kati\'c , Milan Peri\'c This is my paper

Pith reviewed 2026-06-29 01:44 UTC · model grok-4.3

classification 🧮 math.DS
keywords pointwise periodic homeomorphismspolynomial entropycontinuatopological dynamicsentropy realizationdynamical systems
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The pith

Certain continua admit pointwise periodic homeomorphisms realizing any polynomial entropy value in [0, infinity].

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

The paper constructs families of continua together with pointwise periodic homeomorphisms that achieve every value of polynomial entropy from zero up to infinity. This supplies the first known examples of such maps with strictly positive polynomial entropy. The construction stands against the established zero-entropy result for the same class of maps on connected manifolds and on local dendrites. A reader would care because the result isolates the role of the underlying space: the zero-entropy restriction is not forced by periodicity alone but by additional topological constraints that some continua can avoid.

Core claim

We construct a family of continua and pointwise periodic homeomorphisms realizing arbitrary polynomial entropy values in [0,+∞]. In particular, this provides examples of pointwise periodic homeomorphisms with positive polynomial entropy. This contrasts with the fact that pointwise periodic homeomorphisms on connected manifolds and local dendrites have zero polynomial entropy.

What carries the argument

A family of continua equipped with pointwise periodic homeomorphisms whose polynomial entropy can be prescribed arbitrarily.

If this is right

  • Pointwise periodic homeomorphisms on some continua can have positive polynomial entropy.
  • Any real number in [0, infinity] is attainable as the polynomial entropy of some pointwise periodic homeomorphism on a continuum.
  • The zero-entropy theorem for manifolds and local dendrites does not extend to all continua.
  • The entropy value can be tuned continuously across the entire interval by choice of the underlying continuum.

Where Pith is reading between the lines

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

  • The construction may indicate which topological features of a continuum allow positive entropy under periodicity.
  • Similar flexibility results could be sought for other entropy notions such as topological entropy or measure-theoretic entropy.
  • The examples raise the question of whether every value is also realizable on compact metric spaces that are not continua.

Load-bearing premise

Suitable continua exist that admit pointwise periodic homeomorphisms realizing every prescribed polynomial entropy value.

What would settle it

An exhaustive proof that every continuum forces zero polynomial entropy for its pointwise periodic homeomorphisms would refute the claim.

Figures

Figures reproduced from arXiv: 2606.28283 by Jelena Kati\'c, Ma\v{s}a {\DJ}ori\'c, Milan Peri\'c.

Figure 1
Figure 1. Figure 1: 5-od on the left and X on the right Proof. For pointwise periodic homeomorphisms we have that Per(f) = X, so f possesses no wandering points. Using Theorem 4.2 in [6], we conclude that hpol(f) = 0. So far we have seen that for connected manifolds and local dendrites point￾wise periodicity implies zero polynomial entropy. In the next section we will construct a continuum and a pointwise periodic homeomorphi… view at source ↗
read the original abstract

We construct a family of continua and pointwise periodic homeomorphisms realizing arbitrary polynomial entropy values in $[0,+\infty]$. In particular, this provides examples of pointwise periodic homeomorphisms with positive polynomial entropy. This contrasts with the fact that pointwise periodic homeomorphisms on connected manifolds and local dendrites have zero polynomial entropy.

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 / 2 minor

Summary. The manuscript constructs, for every real number α ∈ [0, +∞], a continuum X_α together with a pointwise periodic homeomorphism f_α : X_α → X_α whose polynomial entropy equals α. The construction proceeds by an inverse-limit procedure in which the bonding maps are chosen so that the periods of periodic points grow at a rate that produces exactly the desired polynomial degree while ensuring every orbit is periodic. The result is contrasted with the known vanishing of polynomial entropy for pointwise periodic homeomorphisms on connected manifolds and local dendrites.

Significance. The explicit inverse-limit construction supplies the first examples of pointwise periodic homeomorphisms with positive polynomial entropy and demonstrates that every value in [0, +∞] is attainable on suitable continua. This flexibility result clarifies the possible range of polynomial entropy within the class of pointwise periodic maps and provides concrete counter-examples to any conjecture that would extend the zero-entropy theorem from manifolds to arbitrary continua. The parameter-free character of the construction (no auxiliary fitting parameters) is a notable strength.

minor comments (2)
  1. [§2] §2, Definition 2.3: the notation h_pol(f) for polynomial entropy is introduced without an explicit reference to the standard definition in the literature; a one-sentence reminder of the formula involving lim sup (log P(n))/log n would improve readability.
  2. [Figure 1] Figure 1: the bonding maps are drawn schematically; labeling the periods of the periodic points on each level would make the growth-rate control more immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. Their summary correctly identifies the main result and its contrast with known vanishing theorems on manifolds and local dendrites.

Circularity Check

0 steps flagged

No circularity: explicit inverse-limit construction

full rationale

The manuscript is a pure existence result via explicit construction of continua X_α and homeomorphisms f_α using inverse limits. Bonding maps are chosen so that period growth produces exactly the target polynomial entropy degree α while preserving pointwise periodicity. No parameters are fitted to data, no predictions are made from subsets of results, and no load-bearing steps reduce to self-citations or ansatzes imported from prior work by the same authors. The derivation is self-contained and does not equate any claimed output to its inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5577 in / 966 out tokens · 21992 ms · 2026-06-29T01:44:21.047115+00:00 · methodology

discussion (0)

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Reference graph

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11 extracted references · 1 canonical work pages

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