Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.

I wanted to have a complete bibliography of work done in ergodic theory that has been directly motivated by this conjecture. What I have is this:

1967: Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.

1990: Rudolph, $\times 2 \times 3$ invariant measures and entropy.

2006: Einsiedler and Katok and Lindenstrauss, Invariant measures and the set of exceptions to Littlewood’s conjecture.

2008: Einsiedler and Fish, Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on $\mathbb{T}.$

I would like if you can help me with this list.

  • 1
    $\begingroup$ I'm sure you would like help. A better phrasing would be: "Are there any items you would recommend adding to this list, or additional resources I could use in finding articles motivated by the conjecture?" I would add a reference-request tag, but I would also expect very little contribution. Most likely people will suggest web search, citation indices, and emailing the authors. You will still have a lot of work to do: best not to sound as if you are trying to farm the work out. Gerhard "Appreciates A Good Reference Request" Paseman, 2014.03.26 $\endgroup$ Mar 26, 2014 at 20:08
  • $\begingroup$ Many thanks. I know that Einsiedler and Lindenstrauss know perfectly the answer. $\endgroup$ Mar 26, 2014 at 20:11
  • $\begingroup$ What do you mean by a "continuous" measure? The usual assumption is "without atom". $\endgroup$ Mar 27, 2014 at 10:53

6 Answers 6


Well that will be some lengthy answer.

The first article that was published after the famous disjointness paper is another paper by Hillel called "Intersections of Cantor sets", it's related to the motivating question to the $\times 2,\times 3$ which arise from normality of numbers and fractals. This can be seen as some early work towards the Pallis conjecture, and I'm sure that googling Pallis conjecture will give you a lot of related resources (including Yoccoz's theorem), although they took it to a different direction (actually, I've mentioned Yoccoz's theorem to Hillel only about 2 years ago and he didn't knew about it at the time).

Afterwards, Daniel Berend (a former student of Hillel) proved the higher dimensional analouge of Furstenberg's density theorem - "Multi-Invariant Sets on Tori".

The first substantial work was done by Russel Lyons, who proved a very weak version of the Rudolph-Johnson theorem.

Afterwards came the Rudolph paper, followed by the Rudolph-Johnson paper (doing the co-prime and the rationally independent cases respectively). The papers used the symbolic dynamics mechanisms and they are a bit complicated, a survey of the method appears in the upcoming book by Manfred, vol II of his Springer GTM books (together with Tom Ward).

Afterwards, Host proved his theorem, which is somewhat more of an equidistribution theorem than a measure-classification theorem.

A very nice extension of Host's theorem was proven by Meiri (a former student of Hillel) in his PHD thesis, in a paper called "Entropy and Uniform Distribution of Orbits in T^d".

It's worth mentioning here (although not completely related) that around that time, Katok started working on the Katok-Spatzier paper (which was one of the motivating reasons to the EKL paper).

The (sort of-) analogue to the Rudolph-Johnson theorem by Host's methods was proven by Elon in a paper called "p-adic foliations". Moreover, Elon and Manfred did some work about the analogue of Furstneberg's theorem for general compact abelian groups in a paper called "Rigidty of $\mathbb{Z}^{d}$ actions on tori and solenoides". Elon and Klaus schmidt showed some analogues in the case of non-expansive automorphisms in "INVARIANT SETS AND MEASURES OF NONEXPANSIVE GROUP AUTOMORPHISMS", there you general need to take care about continuity of entropy and such technical considerations.

Then came the famous EKL paper (another worth mentioning paper is Lindenstrauss' "1.5" rigidity paper about the Adelic geodesic flow on $PGL_2$). The EKL paper deals exclusively with the classical settings of homogeneous flows, and not the $S$-arithmetic settings which is needed to Furstenberg's conjecture. I'm not even sure if the full statement is written somewhere, I'm suggesting looking at more recent work of Manfred and Elon (they have somewhat long paper about structure of measures invariant under reductive group action where the group is defined over a local field, this is probably the most complete up-to-date statement of the measure-classification for diagonal actions, although many of the delicate manners are when dealing with positive char. and not in char. zero settings).

Afterwards, the more recent "effective Furstenberg" proven by Bourgain,Lindenstrauss,Michel and Venkatesh. After that, Wang (a student of Elon) proved an "effective Berend" theorem for his PhD thesis.

Warning: Self-promotion, if Benoit did then so do I can. The most recent development is a proof by myself (which is circulating for the last couple of years or so) of a "sparse Furstenberg" theorem, dealing with density of sets such as $\{2^{n}3^{3^{3^{k^17}}}3^{3^{3^{3^{m^{89}}}}}.x\}$, the exact towers involved have some $3$-adic structure, and the proof builds on the work of BLMV, combining the Host-Meiri results.

The recent work by Mike Hochman (some of the papers are in collaboration with Pablo Shmerkin, but in general Mike can generate at-least $2$ proofs by his methods) is different then the above mentioned methods, and is more in the spirit of Furstenberg's papers about intersections of Cantor sets. Mike uses some nice arithmetic combinatorics theorems (work related to the discretized ring conjecture) to get some "structural theorem" about the fractal measures and somehow he can discriminate the parts which contribute to the entropy. This is different than Host's line of attack, which is used in most of the papers since Host's paper.

P.S. a very good source (although a bit dated, not including the effective theroems and Hochman's work) is a survey article by Elon in the FurstenbergFest proceedings called - "Rigidity of multiparameter actions".


The recent landmark paper by Hochman and Shmerkin is relevant as well.


Klaus Schmidt in Vienna wrote (but never published) an extensive set of notes called "$\times$$\beta$, $\times$2, and $\times$3" (the $\beta$ refers to the $\beta$-transformation $x\to \beta x \pmod{1}$ ). The version I have, dated 2009, is about 200 pages, and has a bibliography of 80 items. You could contact him for this document.


This paper by Bourgain, Lindenstrauss, Michel, Venkatesh is relevant.


Some papers relevant to Furstenberg's conjecture (or its version where 2 and 3 are replaced by arbitrary p,q relatively prime-- but your listing of the 2008 paper seems to indicate that you are OK with generalizations) may bear MSC classification in number theory instead of dynamical systems, e.g. this one: MR1348326 (96g:11092) Host, Bernard(F-CNRS-DM) Nombres normaux, entropie, translations. (French. English summary) [Normal numbers, entropy, translations] Israel J. Math. 91 (1995), no. 1-3, 419–428

Theorem 2 in this paper says the following: Let $p$ and $q$ be relatively prime natural numbers and $\mu$ a probability measure on $\mathbb{R}/\mathbb{Z}$ conservative for the group of $p$-adic rational numbers and invariant under the transformations $S:x \to qx$. Then $\mu$ is the Lebesgue measure.

I might be suggesting something obvious here, but a good way to account for related papers would be to open the MathSciNet review of the original paper by Furstenberg (published in 1967, not 1976, by the way)-- MR number MR0213508-- and click on the list of "citations from references" (to be found in the upper right corner) and then go through the list of papers that opens.


Warning : this answer is self-promotion.

In a recent paper I used the differential structure given by optimal transport and computed the (Gâteaux) differential at the point $\lambda$ (=Lebesgue measure) of the actions of $\times_d$ on measures.

I only noticed later (see my habilitation, in French) that this computation shows that the "infinitesimal" version of Furstenberg conjecture is very wrong: there exists an infinite-dimensional subspace of the tangent space at $\lambda$ that is made of vectors invariants under both $\times_2$ and $\times_3$.

Edit: To make things clearer, let me precise a few points.

  • About the differential structure I am considering. In the usual affine structure, an infinitesimal movement (aka tangent vector) is represented by a signed measure, i.e. we record how the density of the moving measure changes at each point. In the differential structure coming from optimal transport, a tangent vector records for each point the direction and speed of the mass lying at this point (a tangent vector on the set of measure can therefore be represented by a vector field on the base space).

  • An interpretation of the failure of infinitesimal Furstenberg conjecture. Optimal transport provides us with a certain metric on the set of probability measures, the "quadratic Wasserstein metric", which is a the heart of the differential structure I allude to. My result says that it is possible to deform $\lambda$ into measures $(\lambda_t)$ such that $\lambda_t$ is at distance $t+o(t)$ from $\lambda$, and both $(\times_2)_\#(\lambda_t)$ and $(\times_3)_\#(\lambda_t)$ are at distance $o(t)$ from $\lambda_t$. In words, one can deform the Lebesgue measure into simultaneously almost-invariant measures. Moreover, there are infinitely many independent ways of doing so.

  • $\begingroup$ what will be the meaning of this result (in a layman terms, for those of us who don't know anything about optimal transport)? It seems (from some googling) that optimal transport (at-least in this dynamical settings) is equivalent to minimizing some function over the ergodic joinings of two systems. Am I correct? Just the statement of your results (as far as my french goes, which is not very far) somehow reminds me of the "unpublished" "Furstenberg-Margulis" conjecutre - large periodic orbits tend to equidistribute. $\endgroup$
    – Asaf
    Mar 28, 2014 at 13:16
  • $\begingroup$ Notice that one cannot expect equidistribtuion of indvidual point orbits, as was already observed by Furstenberg in the disjointness article, where he considers Liouville type numbers in base $6$. $\endgroup$
    – Asaf
    Mar 28, 2014 at 13:17

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