Lifting Kloeckner's infinitesimal counterexamples to the Furstenberg $\times 2,\times 3$ rigidity problem: differentiability of push-forward maps on Wasserstein space and genuine deformations of invariant measures

Furstenberg conjectured in 1967 that Lebesgue measure $\lambda$ is the only atomless probability measure on the circle invariant under both maps $x\mapsto 2x \bmod 1$ and $x\mapsto 3x \bmod 1$; the conjecture remains open. Kloeckner (2013, 2017) proved a strong failure of the first-order analogue of this rigidity: in the differential structure that optimal transport induces on the space of probability measures, the space $E_2\cap E_3$ of tangent vectors at $\lambda$ that are infinitesimally invariant under both push-forward maps is infinite-dimensional, and the space of vectors infinitesimally invariant under all integer multiplications is 2-dimensional. This record poses the questions that work leaves open, as a single coherent objective: does this first-order non-rigidity lift to genuine non-rigidity of the set of atomless $(\times 2,\times 3)$-invariant measures? Concretely: (1) is the push-forward of an expanding circle map Gâteaux-differentiable, in the Wasserstein sense, at any measure that is not absolutely continuous with respect to Lebesgue? (2) does some infinitesimally invariant tangent vector extend to a vector field preserved by both push-forwards, whose integral curve from $\lambda$ would consist of atomless invariant measures — an actual counterexample to Furstenberg's conjecture? (3) the target case of the finitely generated semigroup $\langle 2,3\rangle$, which is the conjecture itself. No progress has been found on any of the three questions; all remain open.

Background

Let $S^1=\mathbb R/\mathbb Z$ denote the circle and, for an integer $d\ge 2$, let $\Phi_d:S^1\to S^1$ be the expanding circle map $\Phi_d(x)=dx \bmod 1$. A probability measure $\mu$ on $S^1$ is invariant under $\Phi_d$ if $\mu=\Phi_{d\#}\mu$, where $\Phi_{d\#}\mu$ is the push-forward measure $\mu\circ\Phi_d^{-1}$; $\mu$ is atomless if $\mu(\{x\})=0$ for every point $x$. Furstenberg conjectured in 1967 that Lebesgue measure $\lambda$ is the only atomless probability measure invariant under both $\Phi_2$ and $\Phi_3$. The statement has remained open for over fifty years. The strongest known partial results are rigidity theorems under additional hypotheses: Rudolph proved the conjecture for invariant measures that are ergodic for the joint $(\Phi_2,\Phi_3)$-action and have positive entropy for one of the two maps (entropy here meaning Kolmogorov–Sinai measure-theoretic entropy); Johnson extended this positive-entropy rigidity to every non-lacunary multiplicative subsemigroup of the integers (a semigroup not contained in a geometric progression $\{\lambda^n\}$); and Einsiedler and Fish proved, without any entropy assumption, that a measure invariant under a 'large' multiplicative semigroup — one of positive lower logarithmic density, for example the full semigroup of all integers $\ge 2$ — and ergodic for the whole semigroup action is either Lebesgue or finitely supported, so that Lebesgue is the only atomless invariant measure for such large semigroups. In particular, the 'large semigroup' cases are rigid, while any counterexample for the finitely generated semigroup $\langle 2,3\rangle=\{2^a3^b\}$ — which is itself non-lacunary but far from 'large' — must have zero entropy for both maps.

Kloeckner approached this rigidity problem through the differential geometry of the space $\mathcal P(S^1)$ of probability measures on the circle, equipped with the quadratic Wasserstein distance $W_2$ of optimal transport (the minimal cost of moving one measure to another with quadratic cost). In the differential structure induced by optimal transport, a tangent vector at an atomless measure $\mu$ is represented by a vector field $v\in L^2(\mu)$, and the displacement of $\mu$ in direction $v$ is the push-forward $(\mathrm{Id}+tv)_\#\mu$. The push-forward map $\Phi_{d\#}:\mathcal P(S^1)\to\mathcal P(S^1)$ is Gâteaux-differentiable at $\lambda$ (and is not Fréchet-differentiable there), and its derivative was computed in Kloeckner's 2013 paper: on the trigonometric basis of $T_\lambda\mathcal P(S^1)$ it annihilates $\cos(2\pi k x)$ and $\sin(2\pi k x)$ when $d\nmid k$, and sends $\cos(2\pi mdx)$ and $\sin(2\pi mdx)$ to $d\cos(2\pi mx)$ and $d\sin(2\pi mx)$. Writing $E_d$ for the $1$-eigenspace of this operator, a natural 'infinitesimal Furstenberg conjecture' would assert $E_2\cap E_3=\{0\}$. Kloeckner's 2017 note proved the opposite in a strong form: $E_2\cap E_3$ is infinite-dimensional, and $\bigcap_{d\ge 2} E_d$ is $2$-dimensional; consequently there exist curves of atomless measures through $\lambda$ that are invariant to first order simultaneously under every map $\Phi_d$. Thus $\lambda$ is very far from first-order rigid — but first-order invariance is not invariance, and the note is explicit that everything beyond this infinitesimal statement is open. For the finitely generated semigroup $\langle 2,3\rangle$ — unlike 'large' semigroups of positive lower logarithmic density, where rigidity holds by the theorem of Einsiedler and Fish — the remainder in the first-order Taylor expansion of $\Phi_{d\#}$ at $\lambda$ can be made uniform over the generators for a fixed simultaneously invariant tangent vector.

Problem Statement

The common objective is to determine whether the first-order non-rigidity of Lebesgue measure $\lambda$ under the joint action of $\Phi_2$ and $\Phi_3$ lifts to genuine non-rigidity of the set of atomless $(\times 2, \times 3)$-invariant measures. The source work poses three linked open questions.

1. Differentiability at singular measures. Is the push-forward map $\Phi_{d\#}:\mathcal P(S^1)\to\mathcal P(S^1)$ Gâteaux-differentiable (differentiable along each tangent direction, i.e. with a directional $o(t)$ remainder) in the Wasserstein differential structure, at any measure that is not absolutely continuous with respect to Lebesgue? This is the prerequisite for even formulating a flow of invariant measures away from $\lambda$, since the derivative is currently known only at Lebesgue measure and, for general $C^2$ expanding maps, at the unique absolutely continuous invariant measure with density bounded and bounded away from zero.

2. Preserved vector fields and invariant curves. Does there exist a nonzero tangent vector $v\in E_2\cap E_3$ that extends to a vector field (on a neighborhood of $\lambda$ in $\mathcal P(S^1)$, or along a curve starting at $\lambda$) preserved by both push-forwards $\Phi_{2\#}$ and $\Phi_{3\#}$, so that the integral curve issued from $\lambda$ consists entirely of atomless measures invariant under both $\Phi_2$ and $\Phi_3$? A positive answer produces a non-Lebesgue atomless $(\times 2, \times 3)$-invariant measure and thereby refutes Furstenberg's conjecture; a proof that no such lift can exist would be the first rigidity statement beyond first order.

3. The target case. Does there exist an atomless probability measure on $S^1$, other than $\lambda$, invariant under every map $\Phi_{2^a3^b}$ of the multiplicative semigroup $\langle 2,3\rangle=\{2^a3^b:a,b\ge 0\}$? This is precisely Furstenberg's 1967 conjecture, the endpoint at which the program aims, and it is open in both directions.

Relations among the items. The umbrella objective is the lift question (2); the three items above are linked sub-claims of unequal strength. Item (1) is a prerequisite for the flow construction in (2) and does not by itself determine the lift. Item (2) is the umbrella statement: a positive answer produces the required non-Lebesgue atomless $(\times 2,\times 3)$-invariant measure and hence settles (3) positively, while a proof that no such lift exists is a rigidity statement beyond first order. Item (3), Furstenberg's conjecture, is the strictly larger endpoint: a negative answer to (3) settles (2) negatively, but a positive answer to (3) need not arise from a lift and so does not by itself settle (2).

References

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