Strongly Independent Matrices and Rigidity of $\times A$-Invariant Measures on $n$-Torus

Enhui Shi; Hanfeng Li; Huichi Huang; Hui Xu

Strongly Independent Matrices and Rigidity of $\times A$-Invariant Measures on $n$-Torus

T0 review · 0 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Strongly independent perturbations force torus-invariant measures to be atomic or Lebesgue.

desk verdict A genuine n-dimensional measure rigidity result with clean proofs; the main theorem appears correct and the paper deserves serious ref. read the letter →

arxiv 1908.02611 v2 pith:X6VPNTAD submitted 2019-08-06 math.DS

Huichi Huang , Hanfeng Li , Enhui Shi , Hui Xu This is my paper · ORCID

classification math.DS MSC 37A0537A2537A4643A0528C1012E05

keywords stronglyindependentmatricesmeasurerigidityn-torusFouriercoefficientsergodictheoryweakmixingstrongFurstenbergconjecture

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

The paper establishes a measure rigidity theorem for endomorphisms of the $n$-torus: an ergodic, weakly mixing, or strongly mixing probability measure that is invariant under the map $x \mapsto Ax$ and also under the maps $x \mapsto (A^j+B_i)x$ for a family of integer matrices $B_1,\dots,B_n$ that are strongly independent over $\mathbb{Q}$ must be either finitely supported (a Dirac measure in the mixing cases) or the Lebesgue measure. To make this work, the authors introduce a purely algebraic notion: a tuple of matrices is strongly independent if no nontrivial linear combination maps any nonzero vector to zero, and they prove a single matrix is strongly independent exactly when its characteristic polynomial is irreducible, yielding abundant examples over $\mathbb{Q}$. A complementary result shows no such tuples exist over algebraically closed fields for $n\ge 2$. The proof runs through a Fourier-coefficient characterization of ergodicity and mixing for $\times A$-invariant measures, which turns the extra invariances into the equations $\hat{\mu}(B_i k)=1$ and then forces the support to be finite.

What carries the argument

The central object is the notion of a strongly independent tuple of matrices over a field $F$: an $n$-tuple $(B_1,\dots,B_n)$ such that for every nonzero column vector $v$, the vectors $B_1v,\dots,B_nv$ are linearly independent, or equivalently every nontrivial linear combination $\sum u_jB_j$ is invertible. For a single matrix $B$, this property holds exactly when the characteristic polynomial of $B$ is irreducible in $F[t]$ (Theorem 1.5), so companion matrices of irreducible integer polynomials provide strongly independent tuples over $\mathbb{Q}$. The other load-bearing mechanism is Theorem 4.1, which characterizes ergodicity, weak mixing, and strong mixing of a $\times A$-invariant measure purely by the limiting behaviour of the Fourier coefficients $\hat{\mu}(A^j k+l)$ along a Følner sequence; this lets the extra invariances be read directly as coefficient identities and, combined with strong independence, forces the finite-support conclusion via Lemma 4.4.

What would settle it

Exhibit an ergodic $\times A$-invariant probability measure on $\mathbb{T}^n$ that is non-atomic, not Lebesgue, and satisfies the extra $\times(A^j+B_i)$-invariance for a strongly independent over $\mathbb{Q}$ tuple for $j$ in a set of Følner upper density one; the theorem says no such measure exists, and the proof predicts one would find $\hat{\mu}(B_i k)\neq 1$ for some nonzero $k$ despite the hypotheses.

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Extended reading notes

Core claim

The central claim is that the one-dimensional rigidity of $\times p$-invariant measures under $\times(p^j+l)$-invariance, previously established in [H], extends to the full $n$-torus once the perturbation $l$ is replaced by a matrix $B_i$ and the tuple $(B_1,\dots,B_n)$ is strongly independent over $\mathbb{Q}$. Stated on its own terms: for a fixed $A\in M_n(\mathbb{Z})$, the family of maps $\times(A^j+B_i)$ for $j$ in a large set (Følner upper density one, positive density, or infinite, depending on the mixing assumption) is enough to rigidify the $\times A$-action. If the invariant measure is not Lebesgue, there is a nonzero frequency $k$ with $\hat{\mu}(k)\neq 0$, and the Fourier characterization of Theorem 4.1 forces $\hat{\mu}(B_i k)=1$ for every $i$. Strong independence then implies the vectors $B_1 k,\dots,B_n k$ are linearly independent, so the support of $\mu$ is contained in a finite set of torus points, giving the dichotomy.

Load-bearing premise

The whole argument rests on the Fourier-coefficient characterization of Theorem 4.1—that Cesàro averages of $\hat{\mu}(A^j k+l)$ over a Følner sequence correctly detect ergodicity, weak mixing, and strong mixing for $\times A$-invariant measures; if that test failed, the proof's step forcing $\hat{\mu}(B_i k)=1$ would collapse.

Editorial extensions

If this is right

Corollary 1.8 follows: there exists an abelian semigroup $S\subseteq M_n(\mathbb{Z})$ for which the Lebesgue measure is the unique non-atomic measure on $\mathbb{T}^n$ invariant under every $\times A$ with $A\in S$ and ergodic under some individual $\times B$.
The Fourier-coefficient characterization of Theorem 4.1 supplies an iff test for mixing of $\times A$-invariant measures that works for any integer matrix $A$ and any Følner sequence, not just for automorphisms.
In the weakly and strongly mixing cases, the finite-support alternative strengthens to a Dirac measure (Lemma 5.1), so the dichotomy in those cases is atomic versus Lebesgue.
Because any matrix with irreducible characteristic polynomial is strongly independent (Theorem 1.5), the theorem provides many explicit families of perturbation matrices for which rigidity holds, requiring no Diophantine or lacunarity conditions.

Reading between the lines

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

The proof of Theorem 1.7 only uses strong independence of the tuple at the single nonzero frequency $k$; one could weaken the hypothesis to 'for every nonzero $k$ with $\hat{\mu}(k)\neq 0$, the vectors $B_1 k,\dots,B_n k$ are linearly independent over $\mathbb{Q}$' and the same finite-support conclusion would hold, a frequency-wise version of the condition.
The non-existence result over algebraically closed fields (Theorem 1.6) suggests the rigidity is inherently arithmetic: over $\mathbb{C}$ there is no algebraic way to force such a rich perturbation family, so the phenomenon depends on the ground field.
A similar strategy might prove rigidity for actions on other compact abelian groups (e.g., solenoids) where Fourier coefficients and exact endomorphisms make sense, using the same strong-independence hypothesis on the Pontryagin dual.

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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 paper does what it says: it proves measure rigidity on T^n for endomorphisms with sparse semigroups of shifts, extending Huang's one-dimensional result and giving a new route into Furstenberg-type questions. The new notion of strongly independent matrices is the right abstraction, and the main theorem is a real step beyond the circle case.

What is actually new: (1) The characterization that a matrix is strongly independent exactly when its characteristic polynomial is irreducible (Theorem 1.5). This is useful and clean. (2) The non-existence of strongly independent tuples over algebraically closed fields (Theorem 1.6). (3) The rigidity theorem itself, Theorem 1.7, which handles ergodic, weakly mixing, and strongly mixing ×A-invariant measures on T^n and concludes either finite support (or a Dirac, in mixing cases) or Lebesgue. This is a substantive result, not a repackaging of known work.

I checked the main line of proof. The Fourier characterization in Section 4 is standard and proved carefully; the L2 approximation steps are compressed but routine. The rigidity argument in Section 5 uses the Fourier identities correctly: the extra invariances force \hat{\mu}(B_i k)=1 for some nonzero k, and the strong independence of the B_i's then forces finite support via Lemma 4.4. That works.

Soft spots: the paper leans on two quoted results, Lemma 4.5 (mean ergodic theorem for amenable semigroups, credited to Bewley) and Lemma 5.1 (atomic weakly mixing implies Dirac, from Huang). Both are standard, and the first is proved in the cited literature. A referee might want the arguments to be more self-contained, but this is not a substantive gap. The density arguments in cases (1)--(3) of Theorem 1.7 are terse; a slightly fuller write-up would help readers less familiar with Følner density arguments. Nothing here looks wrong.

Who this is for: ergodic theorists working on Furstenberg-type measure rigidity, especially people interested in higher-dimensional generalizations. The linear algebra part (Section 3) is also of independent interest. The paper deserves a serious referee; I would accept it for a strong dynamics journal, with minor suggestions for expanding the compressed steps. It does not resolve the full Furstenberg conjecture, but that is not a flaw.

Referee Report

0 major / 7 minor

Summary. The paper introduces the notion of strongly independent n-tuples of matrices over a field F: for every nonzero column vector v, the vectors B1v,...,Bnv are linearly independent. It proves Theorem 1.5, that a nonzero matrix is strongly independent exactly when its characteristic polynomial is irreducible, and Theorem 1.6, that no strongly independent n-tuple exists over an algebraically closed field for n≥2. The main dynamical result is Theorem 1.7: if an ergodic, weakly mixing, or strongly mixing ×A-invariant Borel probability measure on T^n is additionally invariant under ×(A^j+B_i) for all i and for a density-one, positive-density, or infinite set of exponents j, with (B1,...,Bn) strongly independent over Q, then the measure is either finitely supported or Lebesgue; in the weakly and strongly mixing cases it is either Dirac or Lebesgue. The proof proceeds through a Fourier-coefficient characterization of ergodicity, weak mixing, and strong mixing for ×A-invariant measures (Theorem 4.1) and then uses Lemma 4.4 to force finite support from the extra invariances. A corollary produces small abelian semigroups of integer matrices for which Lebesgue measure is the unique non-atomic invariant measure.

Significance. If the results hold, Theorem 1.7 is a substantial higher-dimensional analogue of the first author's one-dimensional rigidity theorem, and the algebraic part is elegant: the equivalence with irreducibility gives an abundant supply of admissible tuples over Q, while Theorem 1.6 shows the obstruction over algebraically closed fields. The Fourier characterization is stated in useful generality, including possibly non-invertible integer matrices and arbitrary Følner sequences, and the rigidity argument is concise and correct. The paper is carefully organized and the main hypotheses are sharp in their density/strength distinctions. The proof is essentially self-contained apart from two quoted lemmas from [H], namely Lemma 4.5 (mean ergodic theorem for amenable semigroups) and Lemma 5.1 (weakly mixing measures with an atom are Dirac); these are used as black boxes rather than as assumptions of the rigidity conclusion, so I do not see circularity. Overall this is a solid contribution with a clean new algebraic condition replacing the scalar non-lacunarity condition.

minor comments (7)

[§4, proof of Theorem 4.1(1)] The passage from the identity (4.5) on trigonometric polynomials to all f,g in L2(T^n,μ) is asserted in one sentence. Please add the standard approximation argument: after proving μ is T_A-invariant, one has ||f∘T_A^j||_2 = ||f||_2 for every j, and this uniform bound allows a density argument. As written the step is correct but too compressed.
[§4, proof of Theorem 4.1(3)] The same compressed density extension appears in the sufficiency direction: from (4.3) for characters to f,g in L2. Please spell out the approximation with the uniform L2 bounds, since this is what turns the Fourier identities into the mixing condition for Borel sets.
[§4, proof of Theorem 4.1(2)] When applying part (1) to μ×μ to prove ergodicity of the product, the text does not explicitly state that μ×μ is invariant under T_{diag(A,A)}; this follows from the T_A-invariance of μ and should be said before invoking part (1).
[§3, proof of Theorem 1.6] The assertion that the homogeneous polynomial f(z)=det[B1z ... Bnz] has a nonzero zero because F is algebraically closed and n≥2 deserves a brief justification: a nonzero homogeneous polynomial of positive degree in at least two variables has a nontrivial projective zero over an algebraically closed field, and if f is identically zero the conclusion is immediate.
[§5, proof of Corollary 1.8] The notation B_i in the definition of the semigroup S is ambiguous. If B_i is meant to be B^i with B0=In, this should be stated explicitly, so that the strongly independent tuple is (B0,B1,...,B^{n-1}).
[§5, proof of Theorem 1.7(1)] After passing to a subsequence of the Følner sequence to make |Fm∩E|/|Fm| tend to 1, the proof should note explicitly that the subsequence is still a Følner sequence, so that Theorem 4.1 can be applied to it.
[Throughout] There are several typographical slips, including 'm ultiplicative' in §1 and 'Adv anced' in reference [Ro]; these should be corrected in a final pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central rigidity conclusion is proved from Fourier identities and an external mean ergodic theorem, not from its own assumptions.

full rationale

The derivation chain of Theorem 1.7 is self-contained against external benchmarks. The extra ×(A^j+B_i)-invariance is converted, via Lemma 4.2, into the Fourier identities μ̂(A^j k + B_i k)= μ̂(k); Theorem 4.1 then turns the ergodic/weakly/strongly mixing hypotheses into the limits that force μ̂(B_i k)=1, and Lemma 4.4 uses the strong-independence hypothesis to conclude finite support. Theorem 4.1 is proved in Section 4 from the mean ergodic theorem for amenable semigroups (Lemma 4.5, explicitly a special case of [B, Theorem 1]) and standard density/extension arguments, not from the rigidity conclusion. No parameter is fitted and no 'prediction' is read back from the target result. The two citations to [H] (Lemmas 4.5 and 5.1) do not raise the score: Lemma 4.5 is externally supported by [B], and Lemma 5.1 is an elementary parameter-free fact used only for the final Dirac refinement in cases (2)-(3), not for the main finite-support-or-Lebesgue conclusion. Thus no circular step is exhibited.

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

The paper is pure mathematics. It introduces one definition and proves results using standard linear algebra and ergodic theory. There are no fitted constants, no empirical data, and no invented entities beyond the named concept, which is a definition rather than a postulate.

assumptions (6)

standard math Hamilton-Cayley theorem: every matrix satisfies its characteristic polynomial.
Used in Proof of Theorem 1.5 to write P_B(B)=f(B)g(B)=0.
standard math Jordan canonical form over the algebraic closure of F.
Used in Proof of Theorem 1.5 to reduce invertibility of f(B) to evaluation at eigenvalues.
standard math Mean ergodic theorem for actions of N along Følner sequences (Bewley [B, Theorem 1]).
Used as Lemma 4.5 to derive the Fourier-coefficient characterizations in Theorem 4.1.
standard math Density of trigonometric polynomials in C(T^n) and L^2(T^n,mu).
Used in Theorem 4.1 to pass from Fourier coefficients to all continuous and L^2 functions.
domain assumption A weakly mixing invariant measure with an atom is a Dirac measure (Lemma 5.1, quoted from [H, Lemma 5.1]).
Used in the final step of Theorem 1.7 to upgrade finite support to Dirac in the weak and strong mixing cases.
standard math A homogeneous polynomial in n variables over an algebraically closed field has a nonzero root for n at least 2.
Used in Proof of Theorem 1.6 to find a nonzero vector annihilating the determinant expression.

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Pith. "Pith review of Strongly Independent Matrices and Rigidity of $\times A$-Invariant Measures on $n$-Torus." pith.science (2026). https://pith.science/paper/X6VPNTAD

@misc{pith2026190802611,
  author       = {Pith},
  title        = {Pith review of: Strongly Independent Matrices and Rigidity of $\times A$-Invariant Measures on $n$-Torus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X6VPNTAD}},
  note         = {Machine review of arXiv:1908.02611}
}

abstract

We introduce the concept of strongly independent matrices over any field, and prove the existence of such matrices for certain fields and the non-existence for algebraically closed fields. Then we apply strongly independent matrices over rational numbers to obtain a measure rigidity result for endomorphisms on $n$-torus.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

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[2]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Berend, Multi-Invariant Sets on Tori, Transactions of the American Mathematical Society, 280 (1983), 509-532

D. Berend, Multi-Invariant Sets on Tori, Transactions of the American Mathematical Society, 280 (1983), 509-532

work page 1983

[2] [2]

T. Bewley. Extension of the Birkhoff and von Neumannn ergodic theorems to semigroup actions. Ann. Inst. H. Poincar\' e Sect. B (N.S.) 7 (1971), 283--291

work page 1971

[3] [3]

Furstenberg

H. Furstenberg. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1--49

work page 1967

[4] [4]

H. Huang. Fourier coefficients of p -invariant measures. J. Mod. Dyn. 11 (2017), 551--562

work page 2017

[5] [5]

A. S. A. Johnson. Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers. Israel J. Math. 77 (1992), 211--240

work page 1992

[6] [6]

Kalinin and A

B. Kalinin and A. Katok. Invariant measures for actions of higher rank abelian groups. Smooth Ergodic Theory and its Applications (Seattle, WA, 1999) , 593–-637, Proc. Sympos. Pure Math., 69 , Amer. Math. Soc., Providence, RI, 2001

work page 1999

[7] [7]

Katok and R

A. Katok and R. J. Spatzier. Invariant measures for higher-rank hyperbolic abelian actions. Ergodic Theory Dynam. Systems 16 (1996), 751--778

work page 1996

[8] [8]

Katok and R

A. Katok and R. J. Spatzier. Corrections to ``Invariant measures for higher-rank hyperbolic abelian actions''. Ergodic Theory Dynam. Systems 18 (1998), 503--507

work page 1998