arXiv:2608.21299v2
Explicit exposure of Haar measure
math.DS37A9922D4037A0537A2037A2537A4637E1043A25
Abstract
We solve the problem, posed some twenty years ago, of explicitly constructing a continuous function whose unique maximizing measure for the doubling map is Lebesgue measure. This is a special case of a more general result: given a nontrivial compact metrizable abelian group and a continuous endomorphism for which normalised Haar measure is ergodic, we explicitly construct a continuous function on the group for which Haar measure is the unique invariant maximizing measure. The function is the uniform limit of a recursively defined sequence of trigonometric polynomials with rational coefficients; every parameter of the recursion is given by an explicit formula, every step is exact, and the rate of convergence is explicit. For expanding circle endomorphisms, ergodic toral endomorphisms, and the one-sided full shift, the construction can moreover be arranged so that the function is given by a Fourier series that converges uniformly in the classical frequency order, each of whose coefficients is rational and computable exactly by a finite procedure. We also show that no continuous function whose unique maximizing measure is Haar measure has absolutely summable Fourier coefficients, so that no such function lies in the Wiener algebra. For expanding maps of the circle, the Fourier coefficients of a continuous function with a unique maximizing measure of full support admit no closed form, in the sense that they satisfy no linear recurrence with polynomial coefficients; this follows from a result of independent interest, that a continuous function on the circle whose Fourier coefficients satisfy such a recurrence is Hölder continuous.
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01Statements2 reported findingsNo statement issues found
The exposure theorems for expanding circle maps, ergodic toral endomorphisms, one-sided full shifts, and general compact metrizable abelian groups are stated with the needed ergodicity and invariance hypotheses. The Wiener-algebra obstruction and the regularity consequence for P-recursive Fourier coefficients are also correctly delimited. No false central statement was found.
Theorems A–DNo issue found
The constructive exposure results have the claimed convergence and computability properties
PDF pages 3–7 and Sections 2–5 · Theorems A–D · arXiv:2608.21299v2
Finite rational detector polynomials separate Haar measure from measures a prescribed weak-star distance away. Long orbit averages flatten each correction, and summable correction sizes give a uniformly convergent exposing function. Relocating each finite frequency block far enough along its dual orbit preserves all invariant integrals, makes the blocks disjoint, and yields the stated ordered Fourier or Walsh convergence with exactly computable rational coefficients.
Theorems E and FNo issue found
Absolute Fourier summability and P-recursiveness are correctly obstructed
PDF pages 7–9 and Sections 6–7 · Theorems E–F · arXiv:2608.21299v2
For a Wiener-algebra function, a Riesz product on a dissociate subsequence of any nontrivial dual orbit produces an invariant competitor unless every nontrivial orbit sum vanishes; the resulting coefficient identity is exactly the weak-coboundary condition. In the circle case, P-recursiveness gives a distributional differential equation, hence Hölder regularity and real analyticity away from finitely many points. The Mañé lemma then turns a full-support maximizing measure for an expanding map into a continuous coboundary, as stated.
02Proofs2 reported findingsNo proof issues found
The detector, flattening, frequency-relocation, Riesz-product, and differential-equation proof chains were checked through their central estimates. The dual action, orbit sums, convergence modes, and regularity hypotheses are used consistently. No incorrect or materially incomplete proof was found.
Sections 2–5No issue found in the reviewed proof
Detector flattening and relocation preserve the optimizing problem
PDF pages 10–31 · Propositions 2.7, 4.1–4.4, and 5.3–5.5 · arXiv:2608.21299v2
The separation gap is uniform on compact sets of measures, while the chosen Birkhoff averages reduce the sup norm without changing integrals against invariant measures. The recursive summability bounds therefore give uniform convergence. Each relocated character differs from its original by composition with an iterate of the endomorphism, so invariant integrals agree term by term; the lacunary placement prevents later blocks from altering earlier coefficients.
Sections 6–7No issue found in the reviewed proof
The obstruction and regularity arguments close without an unstated gap
PDF pages 32–49 · Propositions 6.5 and 7.11 and proofs of Theorems E–F · arXiv:2608.21299v2
Dissociativity makes the prescribed first Fourier coefficients of the Riesz product exact, and orbit averaging makes the competitor invariant; absolute summability justifies exchanging its Fourier series with integration. For the holonomic part, the recurrence is converted to a finite-order distributional ODE, the local singular analysis supplies a power modulus at the finitely many singular points, and compactness patches the local estimates into global Hölder continuity.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.