The Berger-Wang formula for order-preserving homogeneous maps on cones

Aljo\v{s}a Peperko; Brian Lins

The Berger-Wang formula for order-preserving homogeneous maps on cones

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Pith's one-line read The paper proves the Berger–Wang formula for bounded, equicontinuous families of order-preserving homogeneous maps on polyhedral cones: generalized and joint spectral radii coincide.

desk verdict A real extension of the Berger-Wang formula to order-preserving homogeneous maps on polyhedral cones, with a mostly solid proof and only minor presentation gaps. read the letter →

arxiv 2509.02787 v1 pith:QGG5DGZT submitted 2025-09-02 math.FA

Brian Lins , Aljoša Peperko This is my paper · ORCID

classification math.FA MSC 47H0747J1015A8015B48

keywords jointspectralradiusgeneralizedBerger-Wangformulaorder-preservinghomogeneousmapspolyhedralconesconesubradiusBarabanovnorms

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a nonlinear version of the classical Berger–Wang formula. For any bounded, equicontinuous family of order-preserving, positively homogeneous maps on a closed polyhedral cone in a finite-dimensional normed space, the generalized spectral radius (built from the cone spectral radii of single maps) equals the joint spectral radius (the exponential growth rate of norms of long products). The equality matters because it turns a difficult worst-case product-growth problem into an eigenvalue problem for individual maps. The proof advances through the finitely many parts of the cone, using a preorder induced by the family and partial spectral radii, with equicontinuity as the key regularity input. The paper then uses the equality to prove continuity of the joint spectral radius, semigroup boundedness criteria, existence of Barabanov norms, and an extension of spectral subradii to wedges.

What carries the argument

The argument is carried by the decomposition of the polyhedral cone $K$ into its parts — the equivalence classes of nonzero vectors under mutual comparability, $x\sim y$ iff $\alpha x\le y\le \beta x$ for some positive $\alpha,\beta$. Because $K$ is polyhedral, there are finitely many parts, and their closures are the faces. On the set of parts the authors define a preorder $P\ge_{\mathcal A} Q$ iff some composition $f\in\mathcal A^*$ sends a point of $P$ above a point of $Q$. The key estimate is the partial joint spectral radius $\hat r(\mathcal A,P)$, the exponential growth rate of $\|f\|_P$ over $f\in\mathcal A^m$. Lemma 2.4 shows that when $r(\mathcal A)<1$, the limiting direction of any diverging orbit from $P$ lies in a part strictly below $P$; Lemma 2.5 then uses equicontinuity of $\mathcal A^m$ (Lemma 1.4) to show that $\hat r(\mathcal A,P)>r(\mathcal A)$ is impossible for a minimal part. Polyhedrality enters through condition G — for every $x\in K$ and $c<1$, sufficiently close points $y\in K$ satisfy $y\ge cx$ — which upgrades convergence of directions to inequalities $f_k(x)\ge c y$, and through finiteness of the preorder, which makes the minimal-part contradiction valid.

What would settle it

To test the theorem's necessity, compute the two radii for Example 2.7: $\mathcal A=\{f_\lambda\}_{0<\lambda<1}$ on $\mathbb R^2_{\ge0}$ with $f_\lambda(x_1,x_2)=(x_1^\lambda x_2^{1-\lambda},\tfrac12 x_2)$ gives $r(\mathcal A)=\tfrac12$ but $\hat r(\mathcal A)=1$. This family is bounded but not equicontinuous — the slope in the $x_2$-direction blows up as $x_1\to0$ — so it isolates exactly the hypothesis in Theorem 2.1 that must fail for the Berger–Wang equality to break. A genuine counterexample to the theorem would have to be a bounded equicontinuous family on a polyhedral cone with unequal radii, and checking equicontinuity is the gate.

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Extended reading notes

Core claim

The paper's central claim is Theorem 2.1: if $K$ is a closed polyhedral cone in a finite-dimensional normed space and $\mathcal A$ is a bounded, equicontinuous family of order-preserving, homogeneous maps $K\to K$, then $$r(\mathcal A)=\hat r(\mathcal A).$$ The inequality $r(\mathcal A)\le \hat r(\mathcal A)$ is immediate from $r(f)\le\|f\|$. The reverse inequality is proved by contradiction on the preorder of parts of $K$: for a minimal part $P$ with $\hat r(\mathcal A,P)>r(\mathcal A)$, Lemma 2.5 shows equicontinuity forces any unbounded sequence of iterates from a point in $P$ to approach a strictly lower part $Q$, whose growth is already bounded by $\beta<1$ after rescaling to $r(\mathcal A)<1$. This contradiction eliminates all parts, and with them $\hat r(\mathcal A)$. The same circle of ideas yields the paper's corollaries: Hausdorff continuity of the joint spectral radius for compact families of order-preserving homogeneous maps on polyhedral cones; boundedness of the generated semigroup when $r(\mathcal A)=1$ for irreducible subadditive families on $\mathbb R_{\ge0}^n$ or for primitive families on any solid cone; existence of monotone extremal norms and Barabanov norms for irreducible subadditive families; and equality of the generalized and joint spectral subradii for homogeneous maps on wedges.

Load-bearing premise

The load-bearing premise is equicontinuity of the whole family $\mathcal A$: without it a bounded family can contain maps with unbounded one-sided slopes on arbitrarily small scales, and Example 2.7 shows that then the two spectral radii can differ.

Editorial extensions

If this is right

When $r(\mathcal A)<1$, all long products $f\in\mathcal A^m$ have norms decaying exponentially, so the family is uniformly asymptotically stable; this follows from the equality $r(\mathcal A)=\hat r(\mathcal A)$ together with Lemma 1.1.
The joint spectral radius is continuous with respect to Hausdorff convergence of compact families of order-preserving homogeneous maps on a polyhedral cone, so close approximations of a family have close growth rates.
For irreducible subadditive families on $\mathbb R_{\ge0}^n$, $r(\mathcal A)=1$ implies the entire semigroup $\mathcal A^+$ is bounded; for primitive families on any solid cone the same conclusion holds without subadditivity.
Irreducible subadditive families admit monotone extremal norms, and compact ones admit Barabanov norms, yielding Lyapunov functions that characterize stability of the nonlinear inclusion $x(m+1)\in\mathcal A x(m)$.
On any wedge, the generalized and joint spectral subradii coincide, and a discrete inclusion is selectably stable exactly when this common subradius is below 1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

I infer that the proof strategy is likely to extend to non-polyhedral cones whose parts form a well-founded poset under $\ge_{\mathcal A}$ and which satisfy condition G; the paper itself leaves the general non-polyhedral case as an open question.
I infer that for applications such as neural-network stability, the practical message is to check equicontinuity — equivalently, an eventual Lipschitz-type bound on the family — before using the spectral-radius equality as a stability certificate, since Example 2.7 shows a bounded family with kinked maps can evade the formula.
I infer that the equality for spectral subradii on wedges opens a nonlinear route to the mortality problem for inclusions: deciding whether there is a composition product with exponentially decaying norm, and the paper's selectable-stability theorem gives a verifiable criterion.
I infer that Lemma 2.8 suggests subadditivity is a sufficient structural condition to replace equicontinuity; extending that lemma beyond polyhedral cones would yield a broad Berger–Wang formula for subadditive order-preserving homogeneous maps, a testable conjecture.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main theorem is real. Lins and Peperko extend the Berger-Wang formula to bounded equicontinuous families of order-preserving homogeneous maps on polyhedral cones, and they do it without the eigenvector-in-interior condition that the recent Deidda–Guglielmi–Tudisco preprint needs. The proof is a clean induction over the parts of the cone using partial spectral radii. I checked the key step in Lemma 2.5: the reader is right that it is terse, but it is also repairable. If every map in A^m shrank every sufficiently large point of O_m, the orbit would be bounded, so one can pick y_k with a non-contracting A^m step. The rest of the proof goes through.

What is genuinely new: Theorem 2.1 and Example 2.7, which shows equicontinuity is necessary. The boundedness results in Section 4, especially Theorem 4.5 for primitive families on arbitrary solid cones, are useful. Lemma 2.8, showing subadditive families are equicontinuous on polyhedral cones, is a neat observation. Section 5's extremal and Barabanov norms are a solid extension of the matrix theory, and Section 6 cleanly extends spectral subradii to wedges.

Soft spots are minor. Theorem 6.3 invokes [5, Lemma 1] where Fekete's lemma would do, and the wording 'assume' is slightly misleading, but there is no mathematical gap. The non-polyhedral case is left open as Question 2.10; given how heavily the proof uses finitely many parts, that is honest, not a flaw. The bibliographic coverage looks fair: they cite the competing preprint [7] and situate their improvement clearly.

Who this is for: anyone working in nonlinear Perron–Frobenius theory, and people using joint spectral radius bounds for neural networks or switched systems. It deserves a serious referee and, in my view, acceptance. I would ask the authors to expand the proof of Lemma 2.5 a bit and drop the unnecessary citation in Theorem 6.3, but neither is load-bearing.

Referee Report

0 major / 6 minor

Summary. The paper proves a nonlinear Berger-Wang theorem: for a bounded, equicontinuous family of order-preserving, positively homogeneous maps on a closed polyhedral cone in a finite-dimensional normed space, the generalized spectral radius equals the joint spectral radius (Theorem 2.1). The proof proceeds by introducing a partial joint spectral radius on the parts of the cone, defining a preorder on parts determined by the dynamics of the family, and proving the key estimate Lemma 2.5 by contradiction. The paper then gives a continuity result for the joint spectral radius with respect to the Hausdorff metric (Theorem 3.3), studies conditions under which the generated semigroup is bounded when the generalized spectral radius is one (irreducible subadditive families and primitive families, Theorems 4.3 and 4.5), constructs extremal and Barabanov norms for irreducible subadditive families on the standard cone (Theorems 5.1 and 5.2), and extends the notions of joint and generalized spectral subradii to homogeneous maps on wedges (Section 6), with applications to selectable stability. A counterexample (Example 2.7) shows that the equicontinuity hypothesis in Theorem 2.1 cannot be dropped, and the paper leaves open the non-polyhedral equicontinuous case.

Significance. If the results stand, the paper gives a substantial nonlinear generalization of the classical Berger-Wang theorem, replacing linear maps on a vector space by order-preserving homogeneous maps on a cone. The proof strategy via partial spectral radii and induction on the preorder of parts of a polyhedral cone is natural and appears sound. The paper is careful to identify the role of each hypothesis: Example 2.6 distinguishes boundedness from equicontinuity, and Example 2.7 shows that equicontinuity is genuinely needed for the Berger-Wang formula. The boundedness and Barabanov norm results for subadditive and primitive families are useful extensions of the linear theory, and the subradius results answer a natural question in the wedge setting. The paper is self-contained, gives complete proofs of the main claims, and clearly states the open non-polyhedral case. These strengths make the paper a valuable contribution to nonlinear Perron-Frobenius theory and spectral radius theory on cones.

minor comments (6)

[Lemma 2.5] The step after the definition of O_m should be expanded: unboundedness alone does not immediately produce a sequence y_k∈O_m with ∥f(y_k)∥≥∥y_k∥≥k for some f∈A^m. The authors should add the short argument that if every sufficiently large y∈O_m satisfied ∥f(y)∥<∥y∥ for all f∈A^m, then boundedness of A^m would force O_m to be bounded, contradicting the assumption.
[Theorem 6.3] The use of [5, Lemma 1] to 'assume' that r̂_*(A)=β_N^{1/N} and r_*(A)=γ_N^{1/N} is not justified as stated, since such infima need not be attained; the proof should be rewritten with standard ε-approximations. In addition, the equality r_*(A)=liminf_m γ_m^{1/m} should be justified from the property γ_{mn}≤γ_m^n, which is not stated explicitly.
[Corollary 4.6] The reduction to r(A)=1 by scaling should note that primitivity rules out r(A)=0: for any nonzero x there is f∈A^+ with f(x)∈intK, and then f(x)≥cx for some c>0, which forces r(f)≥c and hence r(A)>0.
[Example 2.7] For clarity, the statement about eigenvalues should say 'the only possible cone eigenvalues' and note that the eigenvalue 2^{-m} may have interior eigenvectors as well as the boundary eigenvector e2.
[Theorem 3.3] The application of Theorem 2.1 to the limiting family A should explicitly note that A is bounded and equicontinuous by compactness, so the Berger-Wang equality is available for the limiting family.
[General] There are minor typographical issues, including 'Berger-W ang' in the running header, the spacing in 'Arzel` a-Ascoli', and 'competive' in reference [1].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof chain is self-contained and built on external standard theorems, with no fitted input or self-citation carrying the conclusion.

full rationale

The paper's central result, Theorem 2.1, is derived by a direct mathematical argument. The generalized spectral radius r(A) and joint spectral radius rhat(A) are defined independently (Lemma 1.1 and Lemma 1.2), and the proof of the nontrivial inequality rhat(A) <= r(A) proceeds by an induction over the finitely many parts of a polyhedral cone (Lemma 2.5 and Theorem 2.1). No parameter is fitted to a subset of the data and then renamed as a prediction; no definition smuggles the conclusion in. The cited external results, such as Fekete's lemma, condition G for polyhedral cones, normality of closed cones in finite-dimensional spaces, and standard cone spectral radius theorems, are used as supporting tools and do not themselves assert the Berger-Wang formula for equicontinuous families. The authors' own prior work appears only as background or as part of the general literature on max-algebra spectral radii, not as the load-bearing justification for Theorem 2.1. In Section 6, the extension of spectral subradii to wedges uses [5, Lemma 1] (Czornik), which is an external cited result, and even that invocation is not essential since Fekete's lemma already provides the relevant existence of limits. There are no instances where an equation is asserted to be a prediction while being identical by construction to an input, and no self-citation chain forces the uniqueness or validity of the main formula. The manuscript even identifies its own remaining open problem (Question 2.10) about the non-polyhedral case, which confirms that Theorem 2.1's content is not an artifact of its assumptions. Therefore the correct circularity finding is a non-finding: score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The theorem rests on standard tools from nonlinear Perron-Frobenius theory and functional analysis. No free parameters are fitted and no entities are invented. The main domain assumptions are polyhedrality of the cone and equicontinuity of the family, both explicit in the theorem statement and both used essentially in the proof.

assumptions (6)

standard math Fekete's subadditivity lemma
Used in Lemmas 1.1 and 1.2 to prove convergence of α_m^{1/m} and the equality of the lim sup and sup formulas for the generalized spectral radius.
standard math Closed cones in finite-dimensional normed spaces are normal
Invoked in Lemma 1.3 and Lemma 2.8 to translate order inequalities into norm inequalities, a key step in the partial spectral radius estimates.
domain assumption Polyhedral cones have condition G and finitely many parts
Cited from [3,14]; the proof of Theorem 2.1 and Lemma 2.5 rely on condition G and the finiteness of parts to run the preorder induction and to control the limit directions.
domain assumption Continuity of the cone spectral radius on polyhedral cones
Used in Theorem 3.3 to derive continuity of r(A) with respect to the Hausdorff metric, citing [13, Corollary 4.5].
domain assumption Existence of eigenvectors for continuous order-preserving homogeneous maps on finite-dimensional cones
This background result from [14, Corollary 5.4.2] underpins the definition and basic properties of the cone spectral radius for individual maps.
domain assumption [5, Lemma 1] for spectral subradii
Used in Theorem 6.3 to approximate the infima β_N^{1/N} and γ_N^{1/N}; the proof actually only needs an epsilon-approximation argument, which the text performs.

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Pith. "Pith review of The Berger-Wang formula for order-preserving homogeneous maps on cones." pith.science (2026). https://pith.science/paper/QGG5DGZT

@misc{pith2026250902787,
  author       = {Pith},
  title        = {Pith review of: The Berger-Wang formula for order-preserving homogeneous maps on cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QGG5DGZT}},
  note         = {Machine review of arXiv:2509.02787}
}

abstract

We prove that the joint spectral radius and generalized spectral radius are equal for any bounded, equicontinuous family of order-preserving, homogeneous maps on a polyhedral cone. We also consider conditions which guarantee that the semigroup generated by a family of order-preserving, homogeneous maps is bounded when its generalized spectral radius $r(\mathcal{A}) = 1$. Finally, we extend the notions of joint and generalized spectral subradii to the setting of homogeneous maps on wedges.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [7]

Nonlinear Joint Spectral Radius

P. Deidda, N. Guglielmi, and F. Tudisco. Nonlinear joint spectral radius, 2025. Preprint, https://arxiv.org/abs/2507.11314

work page Pith review arXiv 2025

[2] [1]

Akian, S

M. Akian, S. Gaubert, and L. Marchesini. The competive spectral radius of families of non- expansive mappings, 2024. Preprint, https://arxiv.org/abs/2410.21097

arXiv 2024

[3] [2]

M. A. Berger and Y. Wang. Bounded semigroups of matrices. Linear Algebra Appl., 166:21– 27, 1992

work page 1992

[4] [3]

A. D. Burbanks, R. D. Nussbaum, and C. T. Sparrow. Extension of order-preserving maps on a cone. Proc. Roy. Soc. Edinburgh Sect. A , 133(1):35–59, 2003

work page 2003

[5] [4]

Butkoviˇ c.Max-linear systems: theory and algorithms

P. Butkoviˇ c.Max-linear systems: theory and algorithms . Springer-Verlag, London, 2010

work page 2010

[6] [5]

A. Czornik. On the generalized spectral subradius. Linear Algebra Appl., 407:242–248, 2005

work page 2005

[7] [6]

Daubechies and J

I. Daubechies and J. C. Lagarias. Sets of matrices all infinite products of which converge. Linear Algebra Appl., 161:227–263, 1992

work page 1992

[8] [8]

L. Elsner. The generalized spectral-radius theorem: an analytic-geometric proof. Linear Al- gebra Appl., 220:151–159, 1995

work page 1995