A weak-mixing, zero entropy measure on the 2-shift which gives equal weight to both symbols

Question

I am currently sketching a paper in the general area of symbolic dynamics in which I would like to be able to use the following fact:

Proposition (proposed): there exists a shift-invariant measure $\mu$ on $\{0,1\}^{\mathbb{Z}}$ such that $\mu$ is weak-mixing and has zero entropy with respect to the shift, and such that $$\mu\left(\left\{(x_i)\in \{0,1\}^{\mathbb{Z}} \colon x_0=0\right\}\right)=\mu\left(\left\{(x_i)\in \{0,1\}^{\mathbb{Z}} \colon x_0=1\right\}\right)=\frac{1}{2}.$$ I would like to know if anyone can suggest a reference to an article or textbook which proves that such a measure exists, or failing that if anyone can think of a very quick and crisp proof.

My grounds for believing that the above proposition should be true are that in the set of all shift-invariant probability measures on $\{0,1\}^{\mathbb{Z}}$ equipped with the weak-* topology, there is a dense $G_\delta$ subset all of whose elements are weak-mixing and have zero entropy (see my earlier question for details). It would be somewhat bizarre if this residual set somehow completely missed the one-codimensional affine subspace of measures satisfying the above equation.

(It is not hard to show that there are weak-mixing, zero-entropy measures for which the two quantities in the above equation are arbitrarily close to one half, or to construct weak-mixing measures which satisfy the above equation and have arbitrarily small entropy, but I would like to be able to go the whole distance. I am indifferent to the matter of whether or not $\mu$ is also strong-mixing, but in order to have zero entropy it is well-known that it cannot be Bernoulli or Kolmogorov.)

Edited to add: all of the answers given below have been extremely helpful. After some thought I have decided to accept Tom's answer since in my opinion it most exactly answers the question as specified, but this is not to overlook the fact that Anthony and RW's answers are also very educational and are somewhat broader in their implications.

Answer 1

So with help from Ian and Anthony it looks like my comment above can be turned into an answer. To summarize:

Let S be the substitution on two symbols given by $0\to001, 1\to11100$. Let $x\in\{0,1\}^{\mathbb N}$ be the limit of $S^n(01)$. Let $\Sigma$ be the orbit closure of $x$ under the shift map.

$(\Sigma,\sigma)$ is uniquely ergodic and weak mixing but not strong mixing (Dekking and Keane '83). Let $m$ be the invariant measure. We have that $m[0]=m[1]=\frac{1}{2}$, just by looking at the leading eigenvalue of the matrix (22|13).

Furthermore, primitive substitution systems have zero entropy. I don't know the original reference for this, but much stronger results are known, for example that the complexity function of any primitive substitution has sublinear growth. Apparently this is proved in J.-J. PANSIOT: Complexite des facteurs des mots innis enegendres par morphismes iteres, 1984, or in Queffelec 'substitution dynamical systems'. I read it as a comment on page 5 of Ferenczi 'complexity in sequences and dynamical systems'.

Thus the system $(\{0,1\}^{\mathbb N},\sigma,m)$ satisfies the requirements of your proposition.

Answer 2

To prevent any further beating around the bush let me explain Anthony's answer - as it is in fact much easier than what he actually wrote.

Take any weakly mixing zero entropy transformation $T$ on a probability space $(X,m)$, and take a subset $A\subset X$ with $m(A)=1/2$. Then the symbolic coding of $T$ determined by the partition $(A,X\setminus A)$ satisfies OP's Proposition.

Answer 3

Take a non-atomic weak-mixing measure $\mu_0$ with 0 entropy on two symbols. Equip $X^+=\{0,1\}^{\mathbb N}$ with the lexicographic order and for $z\in X^+$ define $[0,z]=\{x\in X^+\colon x\le z\}$ and $\Phi(z)=\mu_0([0,z])$. Since $\mu_0$ is non-atomic, it's easy to check that $\Phi(z_0z_1\ldots z_n01111\ldots)=\Phi(z_0z_1\ldots z_n10000\ldots)$. In particular, we can define a map $\phi$ from $[0,1]$ to $[0,1]$ by $\phi(t)=\Phi(z(t))$ where $z(t)$ is one of the binary expansions of $t$.

Now notice that $\phi$ is a continuous increasing function with $\phi(0)=0$ and $\phi(1)=1$. Hence there exists $\alpha$ such that $\phi(\alpha)=\frac 12$.

Finally define a map $\Psi$ from $\{0,1\}^{\mathbb Z}$ to itself by $$ \Psi(x)_n=\begin{cases} 0&\text{if $\sum_{k=0}^\infty 2^{-k-1}x_{n+k}<\alpha$;}\\ 1&\text{otherwise.} \end{cases} $$

Now defining $\mu=\mu_0\circ\Psi^{-1}$ gives a weak-mixing measure of 0 entropy with frequency of 1's exactly $\frac 12$.