Getting unique ergodicity from minimality

Question

It is known that minimality does not imply unique ergodicity (Furstenberg example). I ask whether the implication holds in following particular situation:

Suppose $X$ is a compact space, $f:X \to X$ is a minimal and uniquely ergodic homeomorphism. Suppose $p: Y \to X$ is a 2-to-1 covering map, and that $g:Y \to Y$ is a homeomorphism covering $f$, i.e., $p \circ g = f \circ p$. Suppose $g$ is minimal. Does it follow that $g$ is uniquely ergodic?

Trivial example: If $f$ is an irrational rotation of the circle then it is easy to see that so is $g$, and so in this case the question has a positive answer.

A less general but already interesting situation is when the covering is trivial, i.e., $Y = X \times \{0,1\}$. In this case, I think that my question is equivalent to the following coboundary rigidity question:

Let $f$ be as above. Let $\phi:X \to \mathbb{Z}_2$ be a continuous map on the group with two elements. Suppose it is a measurable-coboundary, i.e., there exists a measurable map $\psi:X \to \mathbb{Z}_2$ such that $\phi = \psi \circ f - \psi$ almost everywhere (w.r.t. the unique $f$-invariant probability measure). Does it follow that $\phi$ is a continuous-coboundary? (I.e., can $\psi$ be chosen continuous?)

Rem.: To relate the two questions, notice that $g(x,t)=(f(x),t+\phi(x))$ is uniquely ergodic (resp., minimal) iff $\phi$ is not a measurable-coboundary (resp., not a continuous coboundary).

Answer 1

I think there is a counter-example.

We'll build a subshift on 2 symbols, 0 and 1. Given a word $W$ with symbols 0 and 1, $\overline W$ will denote the word with all symbols flipped. Let $W_0=1$ and $W_{n+1}=W_n^{2^n}\overline{W_n}W_n^{2^n}$.

The orbit closure of $W_\infty$ gives a minimal, but not uniquely ergodic subshift: The orbit closure of $W_\infty$ is the same as the orbit closure of $\overline{W_\infty}$, but the frequency of 1's in $W_\infty$ is greater than $\frac 12$. Let $X$ be the subshift consisting of the orbit closure and consider the map $\Phi\colon X\to \{0,1\}^\mathbb Z$ defined by $\Phi(x)_n=(x_n+x_{n+1})\bmod 2$. Under this map, $W_\infty$ and $\overline{W_\infty}$ map to the same word.

The image is the subshift generated in the following way: Let $V_0=0$ and for any word $v$, let $v^*$ denote $v$ with the last digit flipped. Then define $V_{n+1}=V_n^{2^n-1}V_n^*V_n^*V_n^{2^n}$. Now the image of $\Phi$ applied to a word starting $W_n^k$ or $\overline{W_n}^k$ starts with $V_n^k$ or $V_n^{k-1}V_n^*$ depending on the symbol that follows.

The image shift is therefore the orbit closure of $V_\infty$. This is uniquely ergodic because it's rank 1.

Answer 2

To put Anthony's example in a more general context, if you are familiar with Vershik maps on Bratteli diagrams and dimension groups, then you can find lots of minimal uniquely ergodic actions on Cantor sets that come from the $Z_2$-orbit space of a minimal action action with two ergodic probability measures, with an involution that interchanges the measures.

More explicitly, begin with a Bratteli diagram representing a simple dimension group with unique trace (for example, the 2-adic odometer is represented by $\lim \times 2: Z\to Z$); to be interesting, lots of the multiplicities should exceed $1$, in fact, they should be quite large (which can be arranged by telescoping). Then we can replace the integer multiplicities by characters (not irreducible of course) of the group $Z_2$, and having degrees equalling the multiplicities. Pick a Vershik adic map, e.g., using the left/right ordering.

This yields a new Bratteli diagram, together with a natural action of $Z_2$, and such that the $Z_2$-orbit space is precisely the original action. Except for degenerate cases, this yields a minimal system with a $Z_2$-action, and necessarily a two to one map. Generically, the new minimal system is uniquely ergodic. However, we can easily construct examples where there are two ergodic measures (corresponding to two pure traces on the dimension group).

For example, if we telescope the 2-adic odometer thingy above, so that at the $n$th level, there are $2^n$ edges to the next point, we can realize a character of $Z_2$ whose dimension is $2^n$, corresponding to the matrix $$ A_n:= \left( \begin{array}{cc} 2^n-1 & 1 \\ 1 & 2^n-1 \end{array} \right). $$ This dimension group, $\lim A_n: Z^2 \to Z^2$ is simple and has two pure traces, and there are lots and lots of similar examples. This particular one can be made to sit over the 2-adic odometer by choosing the ordering for the Vershik map appropriately.

This forms part of the classification of actions of $Z_2$ (and other groups) on AF C*-algebras which leave the algebraic direct limit stable. There are a few papers in this area, but I've gone on long enough.