# Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive probability measures on $S$ defined as follows: $$\mu * \nu (A) = \int \int \mathbf{1}_{*^{-1}(A)} (x,y)\ d \nu (y)\ d \mu (x)$$ Here is the question: Is there an idempotent measure in $P(S)$ (i.e. a $\mu$ such that $\mu * \mu = \mu$)? Has this question been considered in the literature?

Here is the motivation: there is a natural way to identify $(S,*)$ with the positive elements of Thompson's group $F$ (elements of $S$ are "rooted ordered binary trees"). It is not hard to show that an idempotent measure is in fact an invariant measure with respect to the action of $F$. Now, I don't expect someone to produce a positive answer (although I will conjecture the even stronger statement that every compact convex $C \subseteq P(S)$ which is $*$-closed contains an idempotent). I am mostly asking if someone sees how to refute the existence of such a measure or if this question has appeared in the literature.

Some further observations: the map $(\mu,\nu) \mapsto \mu * \nu$ is NOT continuous ($\mu \mapsto \mu * \nu$ is, for each $\nu$, but this is about the extent of continuity). Thus the map $\mu \mapsto \mu * \mu$ is not continuous (if it were, we could apply a fixed point theorem...). If one drops the assumption of freeness, then it is possible to find idempotents if any of the following are true:

• $S$ is finite (in this case everything is continuous and so fixed point theorems apply).
• $(S,*)$ is associative (i.e. $S$ is a semigroup) (this is "Ellis's Lemma'').
• $*$ depends only on one argument (again, fixed point theorems apply).

An auxiliary question is to characterize when a monogenic binary system $(S,*)$ satisfies that $P(S)$ contains an idempotent. It should be noted that for associative binary operations, it is possible to find an idempotent ultrafilter (i.e. taking values in $\{0,1\}$) but that this is impossible for the free (non associative) binary system one one generator.

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This is not an answer but perhaps it gives an idea where to look for it. I think that existence of idempotent measures on $\mathbb N$ is the key point in Furstenberg's proof of van der Waerden's theorem and its generalizations and that one can view van der Waerden's theorem as a reformulation of existence of idempotent measures (I may be wrong here). An analog of van der Waerden's theorem is true for the free binary system (groupoid) with $n$ generators: for every decomposition of $A$ into $k$ parts there exists a polynomial $p(x)$ (i.e. a term with one variable) such that all elements $p(x_i), i=1,...,n,$ belong to the same partition class (it is not difficult and was proved by Bespamyatnyh and myself in 1982 but could have been proved by somebody else before). I do not know if it implies existence of idempotent measure, but it may be worthwhile looking at it.