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arxiv: 2512.19430 · v4 · submitted 2025-12-22 · 🧮 math.DS

A survey on the growth rate inequality for sphere endomorphisms

Juliana Xavier This is my paper

Pith reviewed 2026-05-16 20:30 UTC · model grok-4.3

classification 🧮 math.DS
keywords sphere endomorphismsgrowth rate inequalitydynamical systemsopen conjecturestopological degreesurvey
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The pith

Sphere endomorphisms satisfy a growth rate inequality in established cases, though several conjectures remain open.

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

The paper surveys recent results on the growth rate inequality for sphere endomorphisms and reviews the current challenges in proving it in full generality. It collects known cases where the inequality holds along with counterexamples in other settings. A sympathetic reader would care because the inequality would give uniform control on how fast orbits or preimages grow under iteration of any continuous map from the sphere to itself. The survey also gathers a list of open problems and conjectures that would complete the picture if settled. This organizes the state of knowledge for anyone working on sphere dynamics.

Core claim

We survey recent results and current challenges concerning the growth rate inequality for sphere endomorphisms, and present a number of open problems and conjectures arising in this context.

What carries the argument

The growth rate inequality for sphere endomorphisms, which bounds the exponential growth rate of preimages under iteration in terms of the topological degree.

If this is right

  • If the inequality holds for all sphere endomorphisms, topological entropy would be bounded by the logarithm of the degree for every such map.
  • The inequality would supply a uniform way to compare the dynamical complexity of maps across different degrees.
  • Resolving the listed conjectures would classify which sphere endomorphisms achieve equality in the growth rate bound.
  • The survey identifies specific families of maps where the inequality is still unproven, directing future proofs or counterexamples.

Where Pith is reading between the lines

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

  • The same growth-rate control might extend to endomorphisms of higher-dimensional spheres or other compact manifolds.
  • Numerical iteration of random sphere maps could provide quick checks on the remaining conjectures.
  • Links to holomorphic dynamics on the Riemann sphere might yield new cases where the inequality is already known.
  • The open problems could interact with questions about invariant measures and their entropy on spheres.

Load-bearing premise

That the growth rate inequality correctly encodes the essential iterative complexity of continuous maps from the sphere to itself.

What would settle it

An explicit continuous map from the sphere to itself whose preimage growth rate exceeds the bound given by its degree would disprove the general form of the inequality.

Figures

Figures reproduced from arXiv: 2512.19430 by Juliana Xavier.

Figure 1
Figure 1. Figure 1: Julia set for f(z) = z 2 − 0.110 + 0.6557i from the set of critical values and such that f −1(U) ⊂ U. Then f has the rate. Going back to Theorem A, we already pointed out that local connectivity is a technical hypothesis used only to guarantee that external rays land. If one wanted to push the proof forwards, one has to consider the case where external rays land on proper subcontinua of ∂R and try to fish … view at source ↗
read the original abstract

We survey recent results and current challenges concerning the growth rate inequality for sphere endomorphisms, and present a number of open problems and conjectures arising in this context.

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 / 1 minor

Summary. The paper is a survey reviewing recent results and current challenges on the growth rate inequality for sphere endomorphisms in dynamical systems. It also presents open problems and conjectures arising in this context.

Significance. If the coverage is accurate and comprehensive, the survey would be a useful consolidation of the literature for researchers in dynamical systems, particularly those studying endomorphisms of spheres. Highlighting open problems and conjectures could help direct future work in the area.

minor comments (1)
  1. The abstract is concise but could briefly mention one or two key results or the precise statement of the growth rate inequality to better orient readers unfamiliar with the topic.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the survey and for recommending acceptance. We are pleased that the manuscript is viewed as a useful consolidation of results, challenges, and open problems in the area of growth rate inequalities for sphere endomorphisms.

Circularity Check

0 steps flagged

Survey paper with no internal derivations or load-bearing claims

full rationale

This is a survey paper whose abstract and content consist solely of summarizing external results, challenges, open problems, and conjectures on the growth rate inequality for sphere endomorphisms. No new theorems, equations, predictions, or derivations are advanced by the author. There are therefore no load-bearing steps that could reduce by construction to self-defined inputs, fitted parameters renamed as predictions, or self-citation chains. All referenced material is external, and the paper's central claim (accurate summarization of the field) is not subject to the enumerated circularity patterns. This is the expected outcome for a pure survey with no quantitative or deductive content of its own.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Survey paper introduces no new free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5296 in / 831 out tokens · 28927 ms · 2026-05-16T20:30:11.127452+00:00 · methodology

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