arXiv:2608.13876v1

Mixing for Free Semigroup Actions of Blaschke Products on the Circle

Ekaterina Shchetka

math.DSmath.SP37A2537C30

Paper metadata

Original submission: August 14, 2026
Claimed MSC: 37A25, 37C30
arXiv categories: math.DS, math.SP
Paper license: arXiv non-exclusive distribution license ↗
Comments: 35 pages

Abstract

We study mixing at a double exponential rate on analytic observables and ask how much of a map is remembered by its rate of mixing. For finite Blaschke products of the circle with a fixed point in the unit disk, and for the free semigroup actions they generate, we give a complete classification: the rate of mixing (no mixing, exponential, or double exponential) is determined by the multiplier of the generators at that fixed point, the invariant measure being the harmonic measure with a pole there. In the double exponential regime, the exponent equals $\log p$ , where $p$ is the minimal local degree of the generators at the fixed point, and we show that this value is sharp. Consequently, the rate is not rigid: it is not stable under $C^1$ -perturbations and does not imply $C^1$ -conjugacy to affine models. Rigidity holds when the exponent is maximal for the degree: a map of degree $q$ whose exponent attains $\log q$ is Möbius conjugate to an affine model.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The classification of free-semigroup mixing rates by the multipliers at the common interior fixed point, the sharp double-exponential exponent, the stated rigidity and non-stability consequences, and the spectral conclusions are supported. Two local notation defects in the correlation convention and an impossible displayed range for an auxiliary radius have evident corrections and do not alter any mathematical conclusion.

Theorem 1.4 and Corollary 1.5Correct

Classification by the multipliers at the common interior fixed point

Pages 5 and 11–17, 26–28 · Theorem 1.4, Sections 3–4 and 8 · arXiv:2608.13876v1

After simultaneous Möbius normalization, every generator fixes zero and preserves Lebesgue measure. A generator with multiplier of modulus one is a rotation and prevents mixing. If every multiplier has modulus below one, the Schwarz-lemma estimate gives a uniform exponential operator-norm bound for every word. If every multiplier is zero, composition raises the order of vanishing by at least the minimal local degree $p$ at each letter, yielding $\|L^a\|\leq C\exp(-\beta p^{|a|}).$ Conversely, repeating a generator with nonzero multiplier produces a matching exponential lower bound and rules out double-exponential mixing. Section 8 transfers all three alternatives to the harmonic measure at the original common fixed point, with the required analytic-norm comparison.

Theorem 5.1 and Corollary 5.2Correct

Sharpness of the exponent and rigidity at maximal local degree

Pages 18–20 · Section 5 · arXiv:2608.13876v1

For a map with local expansion $\phi(z)=c_pz^p+O(z^{p+1})$ , iteration has leading term $c_p^{(p^n-1)/(p-1)}z^{p^n}.$ The lacunary test observable isolates that coefficient along an infinite subsequence, while the remaining Fourier tail has a strictly smaller double-exponential base. This proves that no exponent larger than $\log p$ is possible. Since the local degree satisfies $p\leq q$ for a degree- $q$ Blaschke product, attaining $\log q$ forces every zero to lie at the fixed point and hence gives the stated Möbius-conjugate affine model.

Propositions 6.1 and 6.2Correct

Failure of stability and nowhere density

Pages 20–22 · Section 6 · arXiv:2608.13876v1

The two explicit degree-three families converge to $z^3$ in the $C^1$ topology, while their local degrees at zero are respectively two and one, so the multiplier criterion places them on opposite sides of double-exponential mixing. Their local degrees also exclude Möbius conjugacy to $z^3$ ; the cited Shub–Sullivan rigidity result upgrades any absolutely continuous conjugacy between the expanding Blaschke products to precisely such a conjugacy. Finally, within the normalized Blaschke products the double-exponential locus is the closed condition $\phi'(0)=0$ , and an arbitrarily small displacement of all but one zero at the origin leaves the normalized class while making the derivative nonzero. This proves nowhere density.

Shub and Sullivan, Expanding endomorphisms of the circle revisited ↗

Theorems 7.1 and 7.2Correct

Spectrum, joint spectral radius, and the quasi-nilpotent algebra

Pages 22–25 · Section 7 · arXiv:2608.13876v1

On the positive and negative Fourier subspaces the compact precomposition operator is triangular with diagonal entries $\lambda^m$ and $\overline{\lambda}^m$ , respectively, which gives the claimed spectrum together with zero. For a word, the multiplier is the product of the letter multipliers. The Berger–Wang formula therefore identifies the joint spectral radius with $\max_j|\lambda_j|$ . When every multiplier vanishes, the word-norm estimate is uniform and double exponential in word length; expanding a power of any zero-constant noncommutative polynomial into words then forces its spectral radius to be zero.

02Proofs4 reported findingsCorrect

The proofs of the classification, sharp exponent, stability consequences, and spectral results are complete. The manuscript has a consistent but local mismatch between its printed correlation convention and the Hermitian pairing used in the calculations, and one auxiliary-radius interval is mistyped. Both are uniquely and harmlessly repairable; neither requires a change to any theorem.

Sections 3–5Correct and complete

Operator bounds, classification, and sharpness

Pages 11–20 · Sections 3–5 · arXiv:2608.13876v1

The Hilbert–Schmidt estimate is reduced to Fourier coefficients of the iterated Blaschke products. In the zero-multiplier case, the order-of-vanishing cutoff makes the remaining geometric sums double exponentially small. In the strict-contraction case, moving the contour to a radius between $\theta$ and one gives an exponentially small factor per letter. Lemma 3.2 supplies the necessary lower bound when a multiplier is nonzero, and the lacunary construction in Theorem 5.1 prevents cancellation from obscuring the leading coefficient. With the two local notation corrections recorded below, every required implication and quantitative comparison closes.

Sections 7–8Correct and complete

Spectral and conjugation arguments

Pages 22–28 · Sections 7–8 · arXiv:2608.13876v1

Compact triangularity determines the one-generator spectrum, and the product formula for derivatives plus Berger–Wang gives the joint spectral radius. The proof for zero-constant operator polynomials correctly controls every term in the $n$ th power by words of length at least $n$ . For a common fixed point away from zero, the single Möbius conjugation transports the action and harmonic measure simultaneously. Lemma 8.1 proves the only nonautomatic point: precomposition sends $H_\theta$ boundedly into an $H_{\theta'}$ corresponding to a slightly smaller annulus.

Correlation conventionTypo

The printed pairing omits the conjugation used throughout the calculations

Pages 4, 13, 18, and 26 · definition of $C_{f,g}$ and later uses · arXiv:2608.13876v1

The definition prints an absolute value around $\int (f\circ\tau(a))g\,d\nu_w-\int f\,d\nu_w\int g\,d\nu_w,$ but the proof then identifies the first integral with the Hermitian $L^2$ inner product, Lemma 4.1 extracts coefficients using $z^{-j}$ , and Theorem 5.1 pairs matching positive Fourier modes. These calculations use the convention $C_{f,g}(a)=\left|\int (f\circ\tau(a))\overline g\,d\nu_w-\left(\int f\,d\nu_w\right)\overline{\left(\int g\,d\nu_w\right)}\right|.$ Inserting the two conjugations makes every displayed computation consistent. Equivalently, one may retain the printed bilinear convention and replace each later $g$ by $\overline g$ ; the function classes and norms are conjugation-invariant. This is a notation-level correction only and leaves all estimates and statements unchanged.

Proof of Theorem 3.1, Part 2Minor formal correction

The initial range printed for the contour radius is empty

Page 17 · opening sentence of the proof in Section 4.2 · arXiv:2608.13876v1

The sentence begins with “for any $0<\theta<1$ and any $1<r<1$ .” Equation (4.2) immediately below chooses the radius in the nonempty interval $r\in(\theta,1)$ , and every preceding and subsequent contour estimate uses exactly that range. Replacing $1<r<1$ by $\theta<r<1$ is the unique local correction and changes no argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.13876v1
Authors listed: Ekaterina Shchetka
Audit date: August 18, 2026

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Mixing for Free Semigroup Actions of Blaschke Products on the Circle

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Classification by the multipliers at the common interior fixed point

Sharpness of the exponent and rigidity at maximal local degree

Failure of stability and nowhere density

Spectrum, joint spectral radius, and the quasi-nilpotent algebra

Operator bounds, classification, and sharpness

Spectral and conjugation arguments

The printed pairing omits the conjugation used throughout the calculations

The initial range printed for the contour radius is empty

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