Burton and Panangaden (2024, published 2026) proved equivalence theorems linking three open ergodic-theoretic problems about measures on the circle invariant under both maps ω↦ω2\omega\mapsto\omega^2 and ω↦ω3\omega\mapsto\omega^3 to statements about Carathéodory functions on the unit disk and tracial states on the full group C∗C^*-algebra of Z2⋉Z[1/6]\mathbb{Z}^2\ltimes\mathbb{Z}[1/6]: Furstenberg's ×(2,3)\times(2,3) conjecture (1967), the periodic approximation question of Levit and Vigdorovich (2024), and the Bourgain–Lindenstrauss periodic equidistribution conjecture. The paper proves only the equivalences; all nine conjectures, grouped into three equivalence classes, remain open. The open core of the problem is to prove or refute each of the three conjectures in any of their nine equivalent formulations.

Background

Write T\mathbb{T} for the unit circle in the complex plane and Prob⁡(T)\operatorname{Prob}(\mathbb{T}) for the Borel probability measures on it. For an integer n≥2n\ge 2, a measure μ\mu is called ×n\times n-invariant if ∫Tf(ωn) dμ(ω)=∫Tf(ω) dμ(ω)\int_{\mathbb{T}} f(\omega^n)\,d\mu(\omega)=\int_{\mathbb{T}} f(\omega)\,d\mu(\omega) for every continuous ff, i.e. μ\mu is invariant under the endomorphism ω↦ωn\omega\mapsto\omega^n; μ\mu is ×(2,3)\times(2,3)-invariant if it is both ×2\times 2- and ×3\times 3-invariant. The set Prob⁡(T;2,3)\operatorname{Prob}(\mathbb{T};2,3) of such measures is a compact convex set, and a ×(2,3)\times(2,3)-invariant measure is called ergodic if it is an extreme point of this convex set. Here λ\lambda denotes Lebesgue (Haar) measure, and the vague topology on measures is the weak-* topology, i.e. convergence tested against continuous functions. In 1967 Furstenberg proved the topological counterpart — the only closed ×(2,3)\times(2,3)-invariant subsets of T\mathbb{T} are finite sets and T\mathbb{T} itself — and conjectured the measure-theoretic analogue: an ergodic ×(2,3)\times(2,3)-invariant measure is either λ\lambda or concentrated on a finite set. The conjecture has been central in ergodic theory for over fifty years as the basic test case for rigidity in number-theoretic dynamics. The only substantial general partial result is due to Rudolph (1990): if an ergodic ×(2,3)\times(2,3)-invariant measure gives one of the two maps positive measure-theoretic entropy, then it is λ\lambda; the zero-entropy regime is untouched. (Note that the intersection conjecture for ×p\times p- and ×q\times q-invariant closed sets, proved by Meng Wu in 2019, concerns sets rather than invariant measures and does not settle the measure conjecture.) Two satellite questions sharpen the same circle of ideas. First, in work on characters of solvable groups and Hilbert–Schmidt stability of groups (approximation of almost-representations by finite-dimensional ones, detectable by traces), Levit and Vigdorovich asked whether every ×(2,3)\times(2,3)-invariant measure is a vague limit of finitely supported ×(2,3)\times(2,3)-invariant ones; the finitely supported invariant measures are exactly those supported on finite unions of finite orbits, and the finite orbits are orbits of roots of unity. Second, Bourgain, in a 2009 paper on the distribution of residues of small multiplicative subgroups of Fp\mathbb{F}_p (whose exponential-sum estimates are the main known partial tool), posed, and Lindenstrauss's 2010 ICM survey publicized as a conjecture, the assertion that any sequence of ergodic finitely supported ×(2,3)\times(2,3)-invariant measures with supports of size tending to infinity must converge vaguely to λ\lambda. Burton and Panangaden recast all three statements in two further languages. Complex-analytically, via the Herglotz correspondence between probability measures on T\mathbb{T} and Carathéodory functions — holomorphic functions on the unit disk D\mathbb{D} with non-negative real part, normalized at the origin, whose Taylor coefficients are the Fourier coefficients of the measure — one obtains a class Cara⁡(D;2,3)\operatorname{Cara}(\mathbb{D};2,3) of Carathéodory functions whose Taylor coefficients satisfy the identities encoding ×(2,3)\times(2,3)-invariance of the measure; under this dictionary finitely supported measures correspond to rational functions and ergodic measures to extreme points. Operator-algebraically, let F2,3=Z2⋉Z[1/6]\mathfrak{F}_{2,3}=\mathbb{Z}^2\ltimes\mathbb{Z}[1/6] be the semidirect product in which the two standard generators of Z2\mathbb{Z}^2 act on the additive group Z[1/6]\mathbb{Z}[1/6] (rationals whose denominator is a power of 6) by multiplication by 2 and by 3, with product (j,k;p)⋆(ℓ,m;q)=(j+ℓ,k+m;p+2j3kq)(j,k;p)\star(\ell,m;q)=(j+\ell,k+m;p+2^j3^kq); the single-map analogue Z⋉Z[1/n]\mathbb{Z}\ltimes\mathbb{Z}[1/n] is a Baumslag–Solitar group. For a countable discrete group GG, the full group C∗C^*-algebra C∗(G)C^*(G) is the universal C∗C^*-algebra generated by unitaries representing GG, and a tracial state is a normalized positive linear functional satisfying the trace condition Φ(ab)=Φ(ba)\Phi(ab)=\Phi(ba); C∗(F2,3)C^*(\mathfrak{F}_{2,3}) carries the regular tracial state Δ\Delta (the coefficient of the identity). A tracial state is conditionally finite-dimensional if, in its GNS representation (the standard construction turning a state into a representation on a Hilbert space), the image of the subalgebra C∗(Z[1/6])C^*(\mathbb{Z}[1/6]) is finite-dimensional, and its conditional dimension cndm⁡\operatorname{cndm} is that finite dimension. Bruce and Scarparo (2024) had already characterized Furstenberg's conjecture tracially: it is equivalent to Δ\Delta being the only faithful extreme tracial state on C∗(Z[1/pq]⋊Z2)C^*(\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2). Burton and Panangaden define an embedding μ↦Φμ\mu\mapsto\Phi^\mu of Prob⁡(T;2,3)\operatorname{Prob}(\mathbb{T};2,3) into the tracial states of C∗(F2,3)C^*(\mathfrak{F}_{2,3}) and prove that the three ergodic conjectures are equivalent, respectively, to rigidity, density, and equidistribution statements for extreme elements of Cara⁡(D;2,3)\operatorname{Cara}(\mathbb{D};2,3) and for tracial states of C∗(F2,3)C^*(\mathfrak{F}_{2,3}). The embedding μ↦Φμ\mu\mapsto\Phi^\mu is not onto in a strong sense: a tracial state is called recoverable if its associated character vanishes off the subgroup Z[1/6]\mathbb{Z}[1/6], equivalently ΦμΦ=Φ\Phi^{\mu_\Phi}=\Phi, and nonrecoverable extreme tracial states exist. The paper proves only quantitative consequences of nonrecoverability, leaving the classification of these states open.

Problem Statement

The common objective is to settle, completely and in full generality, the rigidity, approximation, and equidistribution structure of ×(2,3)\times(2,3)-invariant probability measures on the circle, together with the corresponding structure of the tracial states of C∗(F2,3)C^*(\mathfrak{F}_{2,3}), by proving or refuting each of the following three conjectures (any of their nine equivalent formulations à la Burton–Panangaden is an acceptable target, since the source proves the equivalences). (A) Furstenberg's ×(2,3)\times(2,3) conjecture: every ergodic element of Prob⁡(T;2,3)\operatorname{Prob}(\mathbb{T};2,3) is either λ\lambda or has finite support; equivalently, every extreme element of Cara⁡(D;2,3)\operatorname{Cara}(\mathbb{D};2,3) is rational; equivalently, every extreme tracial state Φ\Phi on C∗(F2,3)C^*(\mathfrak{F}_{2,3}) satisfies Φ=Δ\Phi=\Delta or is conditionally finite-dimensional. (B) Periodic approximation (after Levit and Vigdorovich): every ×(2,3)\times(2,3)-invariant measure on T\mathbb{T} is a vague limit of finitely supported ×(2,3)\times(2,3)-invariant measures; equivalently, every element of Cara⁡(D;2,3)\operatorname{Cara}(\mathbb{D};2,3) is a compact-uniform limit of rational elements; equivalently, every tracial state on C∗(F2,3)C^*(\mathfrak{F}_{2,3}) is a pointwise limit of conditionally finite-dimensional tracial states. (C) Periodic equidistribution (Bourgain–Lindenstrauss): if (μn)(\mu_n) is a sequence of ergodic finitely supported elements of Prob⁡(T;2,3)\operatorname{Prob}(\mathbb{T};2,3) with ∣supp⁡(μn)∣→∞|\operatorname{supp}(\mu_n)|\to\infty, then μn→λ\mu_n\to\lambda vaguely; equivalently, rational extreme elements of Cara⁡(D;2,3)\operatorname{Cara}(\mathbb{D};2,3) whose number of poles diverges converge compact-uniformly to the constant function 1; equivalently, extreme conditionally finite-dimensional tracial states Φn\Phi_n with cndm⁡(Φn)→∞\operatorname{cndm}(\Phi_n)\to\infty converge pointwise to Δ\Delta. In full generality — no entropy hypotheses, no growth restrictions on orbits, no restrictions to subsequences — (A) and (C) each remain open, while (B) follows from (A); no other implication between the three groups is known. A famous-problem warning applies: a proof of only the positive-entropy case, or of any restricted variant (fixed growth regime, particular orbit families, quotient systems), is progress on the program but is not a resolution of (A)–(C).

References

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