Furstenberg's $\times p,\times q$ conjecture: uniqueness of non-atomic $\times p,\times q$-invariant measures on the circle, equivalently uniqueness of $T_p,T_q$-fixed homeomorphisms of $[0,1]$

Let $p,q\ge 2$ be multiplicatively independent integers ($\log p/\log q\notin\mathbb{Q}$) acting on the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ by $x\mapsto px\bmod 1$ and $x\mapsto qx\bmod 1$. Furstenberg's 1967 $\times p,\times q$ conjecture — described in 2024 literature as one of the most fundamental open questions in ergodic theory — asserts that the only non-atomic ('continuous') Borel probability measure on $\mathbb{T}$ that is ergodic and invariant under both maps is the normalized Lebesgue measure. The source paper (Huang, arXiv:1607.02644) reformulates the conjecture equivalently as a functional equation: the only homeomorphism $f$ of $[0,1]$ satisfying $f=T_pf=T_qf$, where $T_ng(x)=\sum_{j=0}^{n-1}\big(g(\tfrac{x+j}{n})-g(\tfrac{j}{n})\big)$, is the identity. Despite substantial partial results — uniqueness under positive entropy (Lyons, Rudolph, Johnson), resolution of the adjacent intersection conjecture (Shmerkin, Wu), and the disproof of a stronger convergence conjecture (Badea–Grivaux) — the conjecture itself, and in particular the zero-entropy ergodic case, remains open.

Background

Multiplication by $n$ on the circle. Write $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ (the unit circle, identified with $[0,1)$) and let $T_n:\mathbb{T}\to\mathbb{T}$ be the expanding map $x\mapsto nx\bmod 1$, which shifts the $n$-ary digit expansion of $x$. A Borel probability measure $\mu$ on $\mathbb{T}$ is $T_n$-invariant if $\mu(E)=\mu(T_n^{-1}E)$ for every Borel set $E$; it is called continuous (or non-atomic) if it gives every single point measure zero, and ergodic for a group or semigroup of maps if every invariant set has measure $0$ or $1$. Each $T_n$ alone has a huge supply of continuous ergodic invariant measures (for example, Bernoulli measures on $n$-ary digit sequences, i.e. measures making the digits independent random variables). Two integers $p,q\ge 2$ are multiplicatively independent if $p^a\ne q^b$ for all positive integers $a,b$, equivalently $\log p/\log q\notin\mathbb{Q}$.

Furstenberg's theorem and conjecture. In his 1967 paper, Furstenberg proved that when $p,q$ are multiplicatively independent, the only infinite closed subset of $\mathbb{T}$ simultaneously invariant under $T_p$ and $T_q$ is $\mathbb{T}$ itself (every irrational $\times p,\times q$ orbit is dense). He conjectured the measure-theoretic analogue: the only continuous (non-atomic) ergodic probability measure invariant under both $T_p$ and $T_q$ is the normalized Lebesgue measure — the $\times p,\times q$ conjecture, with $(p,q)=(2,3)$ as the prototype. The conjecture expresses the heuristic that expansions in multiplicatively independent bases share no structure. It is open; the main partial results all require positive entropy: Lyons (1988) proved the case where $\mu$ has completely positive entropy under $T_p$ (every non-trivial factor system has positive Kolmogorov–Sinai entropy) for coprime $p,q$ — the first place the conjecture appeared in print; Rudolph (1990) weakened this to $\mu$ ergodic under the joint action with positive entropy under one generator, and A. S. A. Johnson (1992) extended it to arbitrary multiplicatively independent pairs. The zero-entropy ergodic case is untouched.

The functional-equation reformulation (Huang, arXiv:1607.02644). For a probability measure $\mu$ on $[0,1)$ define its distribution function $D_\mu(x)=\mu([0,x))$. Huang (arXiv:1607.02644) introduces the bounded linear operator $T_n:C[0,1]\to C[0,1]$ given by $T_ng(x)=\sum_{j=0}^{n-1}\big(g(\tfrac{x+j}{n})-g(\tfrac{j}{n})\big)$ and proves that a continuous measure $\mu$ is $\times n$-invariant if and only if $D_\mu$ is $T_n$-invariant ($T_nD_\mu=D_\mu$); also $T_nT_m=T_{nm}$, so a multiplicative semigroup $\Sigma\subset\mathbb{N}$ acts on $C[0,1]$, and one writes $C_\Sigma[0,1]$ for the linear space of $\Sigma$-invariant functions. Huang then states (as 'Conjecture Cont') that the only homeomorphism $f$ of $[0,1]$ with $f=T_pf=T_qf$ is the identity, and records it as equivalent to Furstenberg's conjecture. The equivalence is sound in both directions for each fixed pair $(p,q)$: a $T_p,T_q$-fixed homeomorphism $f$ defines a continuous $\times p,\times q$-invariant measure through $D_\mu=f$, whose ergodic decomposition cannot contain atomic components (there are only countably many finite rational orbits, and any mixture over them has atoms), so Furstenberg's conjecture forces $f=\mathrm{id}$; conversely, an ergodic non-atomic $\times p,\times q$-invariant measure has infinite, hence full, support by Furstenberg's closed-set rigidity, so its distribution function is a strictly increasing homeomorphism, and Conjecture Cont forces it to be the identity, i.e. the measure is Lebesgue. Huang also proves that $\dim C_\Sigma[0,1]=1$ (the fixed space consists only of multiples of $x$) is a sufficient condition for the conjecture — a stronger statement, since $C_\Sigma[0,1]$ contains non-monotone fixed functions that no measure constrains. As a cautionary example he shows the Cantor function (the distribution function of the middle-thirds Cantor measure) is $T_3$-invariant: singular monotone fixed functions exist, but it is not a homeomorphism and not Lipschitz; determining which regularity assumptions force the identity is part of what makes the functional-equation form hard. Jiang (arXiv:2206.13569) later proved, for coprime $p,q$, that a non-atomic $\times p,\times q$-invariant measure with 'balanced geometry', shown there to be equivalent to a Lipschitz property of one conjugacy, must be Lebesgue — a partial answer on the regularity side.

Problem Statement

Let $p,q\ge 2$ be multiplicatively independent integers ($\log p/\log q\notin\mathbb{Q}$), and write $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ for the circle. Following the source paper, the symbol $T_n$ is used for two related objects: the circle endomorphism $T_n:\mathbb{T}\to\mathbb{T}$, $x\mapsto nx\bmod 1$, and the bounded linear operator $T_n:C[0,1]\to C[0,1]$ on continuous real-valued functions given by

$$T_ng(x)=\sum_{j=0}^{n-1}\Big(g\big(\tfrac{x+j}{n}\big)-g\big(\tfrac{j}{n}\big)\Big).$$

Resolve Furstenberg's $\times p,\times q$ conjecture (1967), equivalently Huang's Conjecture Cont (2016): prove, or refute, that

$$\text{the only non-atomic ergodic Borel probability measure on } \mathbb{T} \text{ invariant under both } T_p \text{ and } T_q \text{ is the normalized Lebesgue measure},$$

and equivalently that

$$\text{the only homeomorphism } f \text{ of } [0,1] \text{ onto } [0,1] \text{ with } f=T_pf=T_qf \text{ is } f(x)=x.$$

Here non-atomic means that no single point has positive measure, and ergodic means that every Borel set invariant under both $T_p$ and $T_q$ has measure $0$ or $1$. The two formulations are equivalent for each fixed pair $(p,q)$: a $T_p,T_q$-fixed homeomorphism $f$ is the distribution function ($D_\mu(x)=\mu([0,x))$) of a non-atomic $\times p,\times q$-invariant measure whose ergodic decomposition contains no atomic components (only countably many finite rational orbits exist, and mixtures over them always have atoms), and conversely an ergodic non-atomic invariant measure has full support by Furstenberg's closed-set rigidity theorem, so its distribution function is such a homeomorphism. The target is the general statement over all multiplicatively independent pairs $p,q\ge 2$, with $(p,q)=(2,3)$ as the prototype; a single explicit counterexample for one multiplicatively independent pair (in either formulation) refutes the conjecture. The auxiliary questions raised in the source — whether $\dim C_{\{p,q\}}[0,1]=1$ (a sufficient condition for the conjecture), and which regularity hypotheses on $T_n$-fixed monotone functions force the identity — are open context, not required deliverables for resolution.

References

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