Timeline for Classification of ergodic measures for circle expanding maps
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 11, 2015 at 20:59 | vote | accept | Benoît Kloeckner | ||
| Mar 1, 2015 at 6:04 | comment | added | Anthony Quas | To be more specific about producing enormous numbers of measures all satisfying your positive entropy condition, given any measure on $\{0,1\ldots,d-1\}^{\mathbb Z}$, take its product with a product measure on $\{0,1\}^{\mathbb Z}$ (where 1 has measure $\epsilon$ and 0 has measure $1-\epsilon$). You can think of the new measure as a measure with $2d$ symbols $\{(0,0),\ldots,(1,d-1)\}$. All of these measures satisfy your concentration condition. As explained before, these measures have any factor with entropy $<\log d$. The randomization part adds $|\epsilon\log\epsilon|$ entropy. | |
| Feb 28, 2015 at 18:40 | comment | added | Anthony Quas | So you can reformulate Furstenberg's conjecture in terms of shift spaces. Indeed this was what Dan Rudolph did when he proved his positive result in the presence of positive entropy. The usefulness of the shift is that it makes it very easy to construct lots of measures using what Kalikow calls the 'monkey method' (see his book with McCutcheon). So for example, it would be easy using this book to construct lots of very bizarre measures that still satisfy your non-concentration condition. | |
| Feb 28, 2015 at 18:11 | comment | added | Benoît Kloeckner | I was aware of the relation with the shift, but wanted to stress Furstenberg's conjecture (which cannot be stated for shifts as they act on different spaces). Universality I didn't know, this sure shows how hopeless it is to imagine a precise classification; but maybe there is something to be hoped when using the metric on the circle; for example maybe one could say something about not-too concentrated measures (e.g. $\mu([a,b])\le C|b-a|^\alpha$ for some $C$ and $\alpha<1$). | |
| Feb 28, 2015 at 17:40 | history | answered | Anthony Quas | CC BY-SA 3.0 |