If $(X,d_X)$ is a compact metric space, $(Y,d_Y)$ some other metric space, and $f : X \to Y$ a function s.t. there are $\alpha > 1$ and $M>0$ with $d_X(f(x),f(y)) \le Md_Y(x,y)^\alpha$, does this imply $f$ is constant? I know it is true when $X$ is a compact interval in $\Bbb{R}$.
@TedShifrin I'm not sure I understand what you're asking. If I assume $X$ is a compact interval in $\Bbb{R}$, then I can prove the theorem. But I don't know how to prove it for a general compact space.
@Ted I have developed a mild annoyance for people who define homotopies without giving the reader any sort of intuition as to how they constructed the formula in the end
Well, if $X = [a,b]$, then take $x,y \in X$ arbitrary but distinct and cut the interval $[x,y]$ into $n$ subintervals of length $\frac{|x-y|}{n}$ for $n \in \Bbb{N}$. Using the triangle inequality $n$ times, we can show $d(f(x),f(y)) < M|x-y|^{\alpha} \frac{1}{n^{\alpha-1}}$. Taking $n \to \infty$ finishes the proof.
I get what you're saying, nowadays I can sorta see what a homotopy is doing geometrically, but when I first started reading about them I didn't have a clue
Specifically, if $X$ is a general compact metric space, take $x,y \in X$ arbitrary but distinct. Then $\overline{B}(x,d(x,y)) \cup \overline{B}(y,d(x,y))$ is compact. I was thinking we could chop it up like I did with $[x,y]$.
@TedShifrin if I have a smooth isotopy $\psi_t$ of a foliated surface $S$ and a fixed leaf $L$ of the foliation, is it immediate that the function $\ell(t) := \psi_t(L)$ of leaves of the foliation is continuous (and smooth)?
I have a foliation which is effectively foliates a plane horizontally. I have one leaf $L$ fixed by picking a point outside of a region $D$ where I deform the plane. Then I deform the plane with $\psi_t$, in particular, outside of $D$, $\psi_t$ is the identity, so it only truly deforms inside $D$.
After $t$-time during the deformation, my fixed leaf $L$ is deformed so it flows out of $D$ into a different leaf (when $t\neq 0$) than it did before. I want to then discuss the continuity of the selection of this new leaf as $t$ varies.
I am unsure how to best phrase this.
And I don't know if I get a "continuous" selection of this new leaf necessarily from the smoothness of $\psi_t$
@anakhro: OK, I just read it. Uniqueness of leaves makes something suspicious there. If the isotopy is the identity outside the set, your leaf $L$ stays the same outside the set. So you can't deform to a different leaf $L'$, which will be different from $L$ outside the set.
So something's not right here, or I'm not understanding.
The isotopy is not leaving the foliation invariant.
But the idea is I want to show that for each leaf segment $L''$ between $L'$ and $\psi(L)$, there is a time $t$ such that $\psi_t(L) = L''$
kind of like an intermediate value theorem thing.
It's awkward phrasing it because I am unsure of how to phrase it formally
In most geometry papers they would just say "it's obvious that...", at least, that seems to be the consensus on this particular lemma I am trying to formalize.
Well yeah, that's sort of what I have right now. It's exactly the x-y plane, and then on the right my leaf segments are called $L_s$ corresponding to their height (off of the disk).
So I guess I do some epsilon delta thing in terms of that parameter s?
