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There any many ergodic, $T$-invariant, non-atomic measures on the space $X = [0,1)$, where $Tx = 2x \pmod 1$ is the doubling map.

My question is: are any such measures also $T^{-1}$-invariant? BYO sigma-algebra.

The question is trivial if we consider a two-sided shift space. But for the one-sided space I can only think of a family of atomic examples:

  • $m_1(x) = \delta_0(x)$ with $\sigma$-algebra $\mathcal B = \left\{\emptyset,\left\{0\right\},X\right\}$, which assigns all the mass to the point $0$,
  • $m_3$ with $\sigma$-algebra $\mathcal B = \left\{\emptyset,\left\{\frac13\right\},\left\{\frac23\right\},\left\{\frac13,\frac23\right\},X\right\}$, which assigns half mass to $\frac13 = 0.0101010101\cdots$ and the other half to $\frac23 = 0.10101010\cdots$, or
  • any measure $m_k$ with $k\geq2$ and $\gcd(k,2)=1$. Where $\mathcal B$ is defined as the smallest $\sigma$-algebra containing $\left\{\frac{j}k : 0 \lt j \lt k\right\}$ and $m_k$ assigns equal mass to all the points: $m_k\left(\frac{j}{k}\right) = \frac{1}{k-1}, 0 < j < k$.

After a bit more thought, the sigma-algebra for such a measure can not contain finite cylinders of measure less than one. For example if $[0]_n = \left\{ x \in X : x_n = 0\right\}$ then $T^n[0]_n = X$ and by $T^{-1}$ invariance of the measure $\mu([0]_n) = \mu(X) = 1$.

This question was posted earlier on [https://math.stackexchange.com/q/1769011/154377].

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  • $\begingroup$ How do you define $\times 1/2$ on the torus? $1/2 \cdot \mathbb{Z}$ is not contained inside $\mathbb{Z}$. You can thnk about extension to solenoids etc but this is pretty much like doing the two-sided extension. $\endgroup$
    – Asaf
    Commented May 3, 2016 at 8:53
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    $\begingroup$ Since $2 \times \left(\frac{1}{2} + \frac{1}{3}\right) = \frac{2}{3} = 2 \times \frac{1}{3}$, the two possible values of $2^{-1} \times \frac{2}{3}$ are $\frac{1}{3}$ and $\frac{5}{6}$. And I would like to define $2^{-1} \times \frac{2}{3}$ as whichever of those two possibilities has non-zero measure. $\endgroup$
    – Daniel Mansfield
    Commented May 3, 2016 at 10:47
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    $\begingroup$ Let's disregard the ergodic + atomic measures (which are supported on periodics, and not very interesting), what would be your definition of $\times 1/2$ for suitable Cantor sets? You should also use more interesting $\sigma$-algebra than just a finite one. Ian Morris have shown some realizations in the Pinsker factor, so let's assume you are outside the Pinsker factor, what happens then? I'm pretty sure you can show that there are generic points (say in a suitable attracting set) with more than one pre-image, hence your question is not well-defined there. $\endgroup$
    – Asaf
    Commented May 3, 2016 at 11:06

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There are very many such measures. In fact, every zero-entropy transformation has a representation as such a measure:

Corollary 4.14.3 in Walters' book states that every zero-entropy measure-preserving transformation of a Lebesgue space is, when restricted to a suitable invariant set of full measure, invertible. By a suitable version of the Jewitt-Krieger Theorem, every zero-entropy ergodic transformation of a Lebesgue space can be represented via measurable isomorphism as a minimal subsystem of the two-sided 2-shift and hence (by the aforementioned corollary) the one-sided 2-shift. By binary coding it follows in particular that every zero-entropy transformation of a Lebesgue space has a representation as an invertible $\times 2$-invariant measure for the doubling map.

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