# Dichotomy Paper

I have a couple questions about the Dichotomy paper of Miller:

1) In Theorem 9 (the Perfect Set Theorem) there is a step that says to “recursively construct a continuous $\psi:2^{\omega}\to 2^{\omega}$ such that $(\varphi\circ\psi)[N_s]\cap(\varphi\circ\psi)[N_t]=\emptyset$ whenever $t,s\in 2^n$ for some $n$. He says that you can do it because $varphi|U$ is non-constant for nonempty open sets.

I’ve tried doing something like the schemes where we assign a clopen set to each node of $2^{<\omega}$ and assign to an $x\in 2^{\omega}$ the intersection of the clopen sets for its inital segments. But I haven't really been able to make it work.

2) In the proof of the silver Dichotomy, we get a perfect clique and the homomorphism ends these to perfectly many equivalence classes. But I am unsure of why there must be a perfect clique.

3) Finally, for Lusin-Novikov, I'm not entirely sure why $G_0$ has to map into one vertical section. I know that each connected component has to go into one section, since we are applying a homomorphism. But the graph isn't connected (right?).

Help with these would be appreciated. Overall the paper looks interesting. I haven't been able to get into the rest of it as much I would have liked. But I think working through these examples will help when it comes to understanding the rest of the paper.

# Image of Fourier Transform

In analysis today, we were talking about the fourier transform and how as a map from $L_2$ to $L_2$ it is onto, but as a map from $L_1$ to $C_0$ it isn’t onto and the range isn’t even closed. Then Dr. Jury said that the range is really hard to describe.

So after class I asked him if “hard to describe” meant “analytic but non-Borel” since the range is analytic by definition. He didn’t know off-hand. I did a quick look online and didn’t find other than people saying that the range is hard to characterize.

So, is there any result saying it is $\Sigma^1_1$ complete or at least non-Borel? I would be interested to know, and to see the proof.

# Closed Graph Theorem in Various Categories

I found this interesting blog post on Terence Tao’s blog about various analogs of the closed graph theorem from functional analysis, which says that a linear function between  Banach spaces $X,Y$ is continuous iff its graph is a (topologically) closed subset of the product $X\times Y$.

# Topological Dynamics

I’m working through the first section on Topological Dynamics.

1) My first questions come towards the end of the proof of Theorem 1.3 on page 119.  We are given that for a unital $C^*$-algebra monomorphism $\pi:C(K)\rightarrow C(L)$, there is a continuous surjection $\Pi:K\rightarrow L$ such that $\pi(f)=f\circ \Pi$.  It says a little later that if $\pi$ is a $G$-map, then $\Pi$ is as well.  In trying to show that, I get so far as $g\cdot \pi(f)(x)=g\cdot(f\circ\Pi)(x)=f(\Pi(g^{-1}x))$ and $\pi(g\cdot f)=(g\cdot f)\circ\Pi(x)=f(g^{-1}\cdot\Pi(x))$.  So these are equal since $\pi$ is a G-map, but to get that $\Pi$ is a G-map from here it looks like we need that $f$ is injective.  It also looks like we need this later when they conclude $\Pi(1_G)=x_0$ from $f(x_0)=f(\Pi(1_G))$, but it doesn’t seem that we can assume that.

2) My next questions come at the top of page 120 concerning the inverse limit of G-flows.  I’ve been getting used to them as I’m reading through this part of the paper, but I am still unsure about how to prove $\lim_{\leftarrow}X_i$ is nonempty and compact.  So I would like some direction on those.  A smaller question later in the paragraph is in the statement of the universal property: if $\phi_i:X\rightarrow X_i$ are homomorphisms and $i\leq j\Rightarrow \phi_i=\phi_{ij}\circ\phi_j\ldots$.  We aren’t given the $\phi_{ij}$ mentioned here, so I wasn’t sure if those were supposed to be the $\pi_{ij}$ or if they were supposed to be introduced along with the $\phi_i$.

3) On page 121, where they start looking at countable discrete G (part B), they say that $S(G)$ is identical to $\beta G$, the space of ultrafilters on G.  Earlier they defined $S(G)$ to be the space of all continuous homomorphisms $\phi:RUC^b(G)\rightarrow \mathbb{C}$ and in this case $RUC^b(G)=l^{\infty}(G)$.  So I’m not really sure how to get from that to the space of ultrafilters.

4) On page 122 (part C) they mention Veech’s theorem about locally compact G acting freely on S(G).  There is a proof in the appendix and I was wondering if this would be a worthwhile proof to look through.

I also had a general question about reading this (or I suppose any) paper.  There are many times in the paper, even the little bit I’ve read, where something is “clear”, or “easily shown”, etc.  These are usually showing something is continuous or is a homomorphism or something like that.  I was wondering if it is a good idea to stop and work through each of these, since most times they aren’t obvious to me, or if it would be better to try and just get through the paper even if some details aren’t completely worked out.

# First Post

I’m starting a blog as I work through the paper Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups by A.S. Kechris, V.G. Pestov and S. Todorcevic.