Timeline for Uniform convergence for pointwise ergodic theorem
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 4, 2020 at 16:25 | comment | added | G. Panel | Can you give a hint for exhibiting such a counter example please? (still considering that the $\mathcal{C}^1$ speed field is non-vanishing on a compact) | |
| Dec 4, 2020 at 6:32 | comment | added | D. Thomine | No, it doesn't (e.g. you can have mutually singular invariant measures with full support). You basically get uniform convergence only in the case where the system has a unique invariant probability measure (unique ergodicity), but that's a pretty specific property. | |
| Dec 3, 2020 at 22:52 | comment | added | G. Panel | If $\mu$ is invariant with support $\text{supp}(\mu)$ and $v$ is still nowhere vanishing, does this convergence happen uniformly toward the choice of $x_0\in\text{supp}(\mu)$? | |
| Dec 3, 2020 at 22:44 | history | edited | G. Panel | CC BY-SA 4.0 |
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| Dec 3, 2020 at 17:26 | history | edited | G. Panel | CC BY-SA 4.0 |
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| Dec 3, 2020 at 17:00 | comment | added | Christian Remling | In any event, the convergence is not uniform in $x_0$. Consider for example a 1D system on $K=[0,1]$, with $v(0)=v(1)=0$ and $v>0$ otherwise. So you move from $x=0$ to $x=1$, and $\mu=\delta_1$, but this will take very long if you start close to $0$. (If you want an example with $v\not=0$, you can make this two-dimensional and approach periodic orbits instead of points.) | |
| Dec 3, 2020 at 16:57 | comment | added | Christian Remling | This is not what the usual ergodic theorem says. Rather, there is a fixed measure $\mu$ (independent of $x_0$), and you have convergence for almost all $x_0$. en.wikipedia.org/wiki/Ergodic_theory#Ergodic_theorems | |
| Dec 2, 2020 at 22:16 | history | asked | G. Panel | CC BY-SA 4.0 |