Timeline for Ergodic, non-atomic measure on the circle which are $\times 2$ and $\times \frac12$ invariant

6 events

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Apr 13, 2017 at 12:19	history	edited	CommunityBot		replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
May 3, 2016 at 11:06	comment	added	Asaf		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.
May 3, 2016 at 10:47	comment	added	Daniel Mansfield		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.
May 3, 2016 at 10:11	answer	added	Ian Morris		timeline score: 5
May 3, 2016 at 8:53	comment	added	Asaf		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.
May 3, 2016 at 6:52	history	asked	Daniel Mansfield	CC BY-SA 3.0