Furstenberg's $\times p,\times q$ conjecture: classification of ergodic invariant measures on the circle, equivalently uniqueness of an irreducible representation of $\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2$

For multiplicatively independent integers $p,q\ge 2$, Furstenberg conjectured (1967) that every ergodic Borel probability measure on the circle $\mathbb{T}$ invariant under both maps $z\mapsto z^p$ and $z\mapsto z^q$ is either finitely supported or the normalized Lebesgue (Haar) measure. The conjecture is open in the zero-entropy regime; the positive-entropy case is the theorem of Rudolph and Johnson. Huang and Wu (2015/2017) recast the conjecture in operator-algebraic terms: via the crossed product duality between ergodic invariant states and irreducible representations, the conjecture is equivalent to the statement that, up to unitary equivalence, there is exactly one irreducible unitary representation of the semidirect product group $\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2$ whose restriction to $\mathbb{Z}^2$ contains the trivial representation and in which no nonzero $\mathbb{Z}^2$-fixed vector is fixed by any nonzero power $\pi(z^k)$, $k\in\mathbb{Z}\setminus\{0\}$, of the canonical generator $z$ of $\mathbb{Z}[1/pq]$ — namely the representation on $L^2(\mathbb{T})$ induced by Lebesgue measure. Later literature (Bruce–Scarparo 2023, Burton–Panangaden 2024/2026, Valente–Yang 2024) supplies further equivalent tracial, complex-analytic, and semigroup-theoretic formulations, all explicitly treating the conjecture as open; a related but distinct Furstenberg conjecture on weak-star convergence of $T_{q^n}\mu$ was disproved by Badea and Grivaux, leaving the classification conjecture itself untouched.

Background

Write $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$ for the unit circle and, for an integer $n\ge 2$, let $T_n:\mathbb{T}\to\mathbb{T}$ be the endomorphism $z\mapsto z^n$ (the $\times n$ map). Two integers $p,q\ge 2$ are called multiplicatively independent if $p^a=q^b$ only when $a=b=0$, equivalently $\log p/\log q$ is irrational. A Borel probability measure $\mu$ on $\mathbb{T}$ is $\times p,\times q$-invariant if $\mu(T_p^{-1}E)=\mu(T_q^{-1}E)=\mu(E)$ for every Borel set $E$, and ergodic if every Borel set with $T_p^{-1}E=E=T_q^{-1}E$ has $\mu$-measure $0$ or $1$. Lebesgue (Haar) measure and measures supported on a finite orbit are the trivial examples. In his 1967 paper on disjointness in ergodic theory, Furstenberg proved that a closed $\times p,\times q$-invariant subset of $\mathbb{T}$ is either finite or all of $\mathbb{T}$, and conjectured the analogous rigidity for measures stated below. The best partial result is the Rudolph–Johnson theorem: if $\mu$ is ergodic $\times p,\times q$-invariant and the measure-theoretic entropy (the asymptotic rate of information production per iterate, in bits per step, of a measure-preserving map) of $T_p$ (equivalently $T_q$) is positive, then $\mu$ is Lebesgue. The zero-entropy case has resisted all attacks for over five decades; the conjecture is the basic test case for the rigidity phenomena believed to govern actions of multiplicative semigroups in number-theoretic dynamics. Huang and Wu recast the problem through crossed products. If a discrete group $\Gamma$ acts by automorphisms on a compact Hausdorff space $X$, the crossed product $C^*$-algebra $C(X)\rtimes\Gamma$ is the universal $C^*$-algebra generated by a copy of the continuous functions $C(X)$ and a unitary representation of $\Gamma$ implementing the action. For a unitary representation $\pi$, write $H_\Gamma=\{x:\pi(\gamma)x=x\ \text{for all}\ \gamma\in\Gamma\}$ for the subspace of $\Gamma$-fixed vectors, so $H_\Gamma\ne 0$ means the restriction of $\pi$ to $\Gamma$ contains the trivial representation. Huang–Wu prove (Theorem 3.10) that for any irreducible representation $\pi$ of $C(X)\rtimes\Gamma$ one has $\dim H_\Gamma\le 1$, and that the irreducible representations with $H_\Gamma\ne 0$ correspond bijectively, up to unitary equivalence, to ergodic $\Gamma$-invariant regular Borel probability measures on $X$. Specializing: the Pontryagin dual (the compact group of all characters, with the topology of pointwise convergence) of the discrete group $\mathbb{Z}[1/pq]$ is the $pq$-adic solenoid $S_{pq}$, on which $\mathbb{Z}^2$ acts through the automorphisms $g\mapsto pg$ and $g\mapsto qg$ of $\mathbb{Z}[1/pq]$; restriction of measures from $S_{pq}$ to $\mathbb{T}\subset S_{pq}$ is a homeomorphism of the corresponding spaces of ergodic $\times p,\times q$-invariant measures (their Proposition 4.1). Since $C^*(\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2)\cong C(S_{pq})\rtimes\mathbb{Z}^2$, ergodic $\times p,\times q$-invariant measures on $\mathbb{T}$ are thereby encoded by irreducible unitary representations of the (amenable, solvable) group $\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2$ whose $\mathbb{Z}^2$-fixed space is nonzero. Their Theorem 4.2 shows the irreducible representations induced by finitely supported ergodic measures are exactly those additionally having some nonzero $N\in\mathbb{Z}$ with $\pi(z^N)$ acting trivially on $H_{\mathbb{Z}^2}$, where $z$ denotes the canonical generator (the element $1$) of $\mathbb{Z}[1/pq]$; and their Corollary 4.3 states that Furstenberg's conjecture holds if and only if there is a unique irreducible unitary representation with $H_{\mathbb{Z}^2}\ne 0$ and with no nonzero vector of $H_{\mathbb{Z}^2}$ fixed by any $\pi(z^k)$, $k\in\mathbb{Z}\setminus\{0\}$ — the one induced by Lebesgue measure. Subsequent equivalent reformulations include the tracial characterization of Bruce–Scarparo (the conjecture holds iff the canonical trace is the only faithful extreme tracial state on $C^*(\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2)$, the full group $C^*$-algebra, i.e. the completion of the group algebra for its universal norm) and the complex-analytic and tracial formulations of Burton–Panangaden.

Problem Statement

Let $p,q\ge 2$ be multiplicatively independent integers. Furstenberg's $\times p,\times q$ conjecture (1967) asks: is every ergodic Borel probability measure on the circle $\mathbb{T}$ that is invariant under both $T_p:z\mapsto z^p$ and $T_q:z\mapsto z^q$ either finitely supported or the normalized Lebesgue measure? Equivalently in the derived representation-theoretic form of Huang–Wu (Corollary 4.3): writing $z$ for the canonical generator (the element $1$) of $\mathbb{Z}[1/pq]$ and letting $\mathbb{Z}^2$ act on $\mathbb{Z}[1/pq]$ by $(m,n)\cdot g=p^m q^n g$ to form the semidirect product $\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2$, the conjecture is true if and only if there is, up to unitary equivalence, exactly one irreducible unitary representation $\pi$ of $\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2$ such that (i) the fixed subspace $H_{\mathbb{Z}^2}\ne 0$ (the restriction of $\pi$ to $\mathbb{Z}^2$ contains the trivial representation), and (ii) $\pi(z^k)x\ne x$ for every nonzero integer $k$ and every nonzero $x\in H_{\mathbb{Z}^2}$ — namely the representation on $L^2(\mathbb{T})$ induced by Lebesgue measure. A complete answer resolves the conjecture in either direction: a proof that no ergodic zero-entropy $\times p,\times q$-invariant measure other than the finitely supported and Lebesgue ones exists (equivalently, the uniqueness statement above), or an explicit ergodic, non-atomic, non-Lebesgue $\times p,\times q$-invariant measure on $\mathbb{T}$ (equivalently, an irreducible representation of $\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2$ satisfying (i) and (ii) but not induced by Lebesgue measure).

References

1. Hillel Furstenberg, 'Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation', Mathematical Systems Theory 1 (1967), 1-49; DOI 10.1007/bf01692494; https://doi.org/10.1007/bf01692494
2. Daniel Rudolph, '×2 and ×3 invariant measures and entropy', Ergodic Theory and Dynamical Systems 10 (1990), 395-406; DOI 10.1017/s0143385700005629; https://doi.org/10.1017/s0143385700005629
3. Aimee S. A. Johnson, 'Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers', Israel Journal of Mathematics 77 (1992), 211-240; DOI 10.1007/bf02808018; https://doi.org/10.1007/bf02808018
4. Huichi Huang, Jianchao Wu, 'Ergodic invariant states and irreducible representations of crossed product C*-algebras', Journal of Operator Theory 78 (2017), 159–172; DOI 10.7900/jot.2016jun11.2141; https://doi.org/10.7900/jot.2016jun11.2141 (preprint arXiv:1505.06633, https://arxiv.org/abs/1505.06633)
5. Xin Ma, 'Invariant ergodic measures and the classification of crossed product C*-algebras', Journal of Functional Analysis 276 (2019), 1276–1293; DOI 10.1016/j.jfa.2018.06.013; https://doi.org/10.1016/j.jfa.2018.06.013
6. Chris Bruce, Eduardo Scarparo, 'A tracial characterization of Furstenberg's ×p,×q conjecture', Canadian Mathematical Bulletin 67 (2024), no. 1, 244-256 (published online September 2023); DOI 10.4153/S0008439523000693; https://doi.org/10.4153/S0008439523000693 (preprint arXiv:2304.04456, https://arxiv.org/abs/2304.04456)
7. Catalin Badea, Sophie Grivaux, 'Around Furstenberg's times p, times q conjecture: times p-invariant measures with some large Fourier coefficients', preprint (2023, revised 2024); arXiv:2303.01089; https://arxiv.org/abs/2303.01089
8. Matan Tal, 'Furstenberg's Times 2, Times 3 Conjecture (a Short Survey)', preprint (2021, revised through February 2026); arXiv:2110.05989; https://arxiv.org/abs/2110.05989
9. Robert Valente, Dilian Yang, 'Semigroups of self-similar actions and higher rank Baumslag-Solitar semigroups', Proceedings of the Royal Society of Edinburgh Section A: Mathematics (2025); DOI 10.1017/prm.2025.10053; https://doi.org/10.1017/prm.2025.10053 (preprint arXiv:2405.07062, https://arxiv.org/abs/2405.07062)
10. Peter Burton, Jane Panangaden, 'Formulations of Furstenberg's ×2×3 conjecture in complex analysis and operator algebras', Annals of Functional Analysis (2026); DOI 10.1007/s43034-026-00512-1; https://doi.org/10.1007/s43034-026-00512-1 (preprint arXiv:2410.22701, https://arxiv.org/abs/2410.22701)