arXiv:2506.01518v2
Typical Uniqueness in Ergodic Optimization
math.DS37A9937B0237D2037F05
Abstract
For ergodic optimization on any topological dynamical system, with real-valued potential function belonging to any separable Banach space of continuous functions, we show that the -maximizing measure is typically unique, in the strong sense that a countable collection of hypersurfaces contains the exceptional set of those with non-unique maximizing measure. This strengthens previous results asserting that the uniqueness set is both residual and prevalent.
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Detailed mathematical audit
01Statements2 reported findingsNo statement issues found
The residual, prevalent, countable-hypersurface, and Lipschitz-sparse uniqueness results are correct for the respective topological vector spaces of potentials.
Theorems 1, 2, and 4No issue found
Unique maximizing measures are topologically and probabilistically typical
PDF pages 3–7 · arXiv:2506.01518v2
Upper semicontinuity of the maximizing-measure multifunction identifies its continuity points with unique maximizers. In a separable Banach space, uniqueness is equivalent to Gateaux differentiability of the convex maximum-average functional, so Zajíček's theorem gives the countable delta-convex hypersurface cover. Christensen's theorem gives the corresponding prevalence result for separable Frechet spaces.
Proposition 3 and Theorem 5No issue found
The exceptional set is still sparse in the countable-extreme-point and Lipschitz settings
PDF pages 6–9 · arXiv:2506.01518v2
With countably many extreme invariant measures, nonuniqueness forces equality of two distinct continuous linear functionals and hence lies in a countable union of hyperplanes. In Lip(X), a countable family of separated Wasserstein ball pairs covers every nonunique case, and each equality locus is a Lipschitz graph over a hyperplane.
02Proofs3 reported findingsNo proof issues found
The proofs correctly connect multifunction continuity, differentiability of the support functional, and Wasserstein separation. The needed continuity and density arguments are supplied.
Proof of Theorem 1No issue found in the reviewed proof
Continuity of the argmax map is equivalent to uniqueness
PDF pages 3–4 · arXiv:2506.01518v2
Closedness of the graph follows by passing the maximizing inequality through convergence in V and weak-star convergence of measures. At a singleton value, upper semicontinuity gives lower semicontinuity. With two maximizers, density of V in C(X) supplies a separating perturbation that destroys lower semicontinuity.
Proof of Theorem 2No issue found in the reviewed proof
Directional derivatives detect exactly the maximizing measures
PDF pages 4–5 · arXiv:2506.01518v2
The two-sided difference-quotient bounds squeeze every directional derivative to integration against the unique maximizing measure. Conversely, two maximizing measures can be separated by a function in V, giving unequal one-sided derivatives. The resulting derivative is a continuous linear functional, so the invoked differentiability theorem applies exactly to the exceptional set.
Appendix, proof of Theorem 5No issue found in the reviewed proof
Wasserstein-separated balls produce Lipschitz hypersurfaces
PDF pages 7–9 · arXiv:2506.01518v2
Every pair of distinct maximizing measures can be captured by disjoint closed balls from a countable dense family. A 1-Lipschitz separator creates a uniform positive slope gap between the two restricted support functions. Hence their difference has one zero on each affine line, and the zero varies Lipschitz-continuously over the complementary hyperplane.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.