1
$\begingroup$

In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?

Further details: How can one use such construction to build a measure on the group of homeomorphisms of $\mathbb T^2$? I would like these properties to hold for the measure: (1) A measure zero set has empty interior; (2) Every subset of a measure zero set has measure zero too; (3) Countable union of measure zero sets has measure zero too; (4) quasi-invariance in some sense. See A measure on the group of homeomorphisms of $\mathbb T^2$ for this question specifically.

Further context: Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$, in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as \begin{equation} h(t)=\tau ([0,t))/\tau ([0,1) )%\qquad \mbox{for}\;\; \theta\in [0,1) \label{h1} \end{equation} Specifically they take $\tau$ formally proportional to $ e^{\beta X(t)}dt$ where $\beta\geq 0$ and $X$ is the Gaussian Free Field on the circle i.e. the random field $X$ with covariance $\mathbb E\, X(t)X(t')=-\log |e^{2\pi it}-e^{2\pi it'}|$ (see within the link and https://arxiv.org/abs/0909.1003 for all the rigorous details).

$\endgroup$
6
  • $\begingroup$ Just consider the "product" $(x,y)\mapsto (f(x),g(y))$ of two such "random" homeomorphisms. You need to require something on the resulting probability measure on Homeo of the torus to make the question nontrivial. $\endgroup$
    – YCor
    Commented Sep 4, 2022 at 7:05
  • $\begingroup$ @YCor That's actually another question that i have: how does one use these random homeomorphisms to build a measure on Homeo? For such measure, ideally, I would like these properties to hold: (1) A measure zero set has empty interior; (2) Every subset of a measure zero set has measure zero too; (3) Countable union of measure zero sets has measure zero too; (4) quasi-invariance in some sense $\endgroup$
    – user490373
    Commented Sep 4, 2022 at 7:42
  • $\begingroup$ @user490373 Their motivation seems to be to exhibit a family of homeomorphisms for which the welding problem has an (essentially) unique solution and to (heuristically) link them to SLE. Neither of these has a higher-dimensional analogue as far as I know. $\endgroup$ Commented Sep 4, 2022 at 19:40
  • $\begingroup$ @MartinHairer I see, thanks. Actually, I'm only interested in building a measure on the group of homeomorphisms of the torus $\mathbb T^2$ and generalizing that work was the first idea that came to mind. But do you see an alternative option to build such measure? Ideally, it should satisfy the following properties: (1) A measure zero set has empty interior; (2) Every subset of a measure zero set has measure zero too; (3) Countable union of measure zero sets has measure zero too; (4) quasi-invariance in some sense $\endgroup$
    – user490373
    Commented Sep 5, 2022 at 6:36
  • $\begingroup$ I conjecture that if you compose many random translations with random homeomorphisms whose support is a small neighborhood of the identity, this composition will converge in distribution to something like a normal distribution on homeomorphisms. You can test whether this version of the central limit theorem is plausible using the Kolmogorov-Smirnov test on some (one real) invariant of your random homeomorphism. $\endgroup$ Commented Sep 10, 2022 at 15:55

0

You must log in to answer this question.