Timeline for Equidistribution on $\mathrm{SU}_2$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 23 at 6:50 | vote | accept | Local | ||
| Feb 23 at 6:50 | vote | accept | Local | ||
| Feb 23 at 6:50 | |||||
| Feb 23 at 3:18 | comment | added | Local | @MikaeldelaSalle Thanks for your comment, dense image looks like a necessary condition for equidistribution, but how to prove the sufficiency? | |
| Feb 22 at 14:03 | comment | added | Mikael de la Salle | The soft argument I provide does not give an a priori speed of convergence, so Bourgain and Gamburd's result gives much more under suitable assumptions of $\rho$ (but I have to admit that I do not see how your argument does, without any assumption on the modulus of uniform continuity of $f$). | |
| Feb 22 at 13:57 | comment | added | Mikael de la Salle | Correct, but it is a much more elementary fact that given $\rho$, the convergence holds if (and only if) $\rho$ has dense image. Indeed, your argument is that $A$, the average of the generators and their inverses acting on $L_2(\mathrm{SU}_2)$ has spectrum contained in $[-1+\varepsilon,1-\varepsilon]\cup\{1\}$ with simple eigenvalue $1$, and therefore $A^n$ converges in norm to the projection on the eigenspace for eigenvalue $1$. But if you only want convergence pointwise, all you need is that $A$ is self-adjoint of norm $\leq 1$ and $-1$ is not an eigenvalue. | |
| Feb 22 at 8:55 | history | answered | Lucas Kaufmann | CC BY-SA 4.0 |